This post has potentially manipulated dice roll results.
BearElspeth appears in a flurry of teeth and claws, ripping and shredding and tearing at the warrior. It's a bit of a blur, and there's no cry from the warrior. But at the end, his form - or what's left of it - lies on the cavern floor, lifeless.
The drake, provoked by the frenzy in front of it, bites at Nate while swiping its tail at BearElspeth:
Bite attack on Nate: Attack: 21 Damage: 5
Tail attack on Elspeth: Attack: 8 Damage: 9
Both attacks hit, but Nate, you feel the jaws close on your leg and the pain is intense - as the drake withdraws, you see your blood covering its mouth and a horrible forked tongue flickering out to taste it.
Follow-up question - would Soar be able to grab the warrior's leg, pick him up to swing the rest of his form at the drake with the aim of knocking the drake off into the pit?
The drake takes its first injury, and turns its head to snap at Jalt.
The kobolds seem taken aback by the death of the mighty warrior who led them, but they keep their distance and attack you all again:
On Soar: Attack: 15 Damage: 3
On Nate: Attack: 5 Damage: 3
On Jalt: Attack: 14 Damage: 5
On Elspeth: Attack: 17 Damage: 5
This time, only Elspeth takes a rock to the face.
Soar and Nate are up.
(What you're describing for Soar is possible. Essentially it would be making a shove attack on the drake; using the warrior to achieve that is part flavour, but also using your environment in a creative way - I might give advantage or disadvantage depending on whether I felt it was easier to push the drake using the large dead body than just pushing. However, I would warn you that the pit isn't all that deep.)
It seems the Drake has clipped the femoral artery in Nate's leg when it took a bite, and Nate is bleeding... bad.
"Ahhhhhhh!!!! GUYS!!! IT'S OUR LAST CHANCE! PLEASE!"
Nate tries to defend himself the best he can by jabbing the Drake with his 2 daggers, basically trying to keep the beast away from him. (he doesn't know what Soar is planning yet, of course.)
This post has potentially manipulated dice roll results.
Soar looks to Nate, very bloodied again, and Jalt, taking big hits but still standing, and says, "Fall back! We'll cover you!"
She looks at the warrior's body, thinking about trying to use it as a cudgel to knock the drake into the pit, but tossing aside that idea. She rushes at the drake, attempting to shove it into the pit, to buy them some time and remove the immediate threat from the board. If she can get some help from Big Brown Bear Elspeth, she will happily accept it.
Soar athletics check with help (with no help, ignore second roll): 12)
(Didn't roll with advantage... here's the second roll if Soar get advantage: 22)
This post has potentially manipulated dice roll results.
Drake's Athletics check: 11
Though Soar's shove is clumsy, the drake is taken completely by surprise; it tries to swivel its tail, but hesitating between using it to keep its balance and attacking the paladin, the creature is knocked down fifteen feet onto the lower level, taking 3 damage from the fall.
Round Four Warrior Elspeth Drake Jalt Kobolds 1, 2, 3 and 4 Nate and Soar
This post has potentially manipulated dice roll results.
Elspeth shoots Soar a grateful look - which doesn't quite translate to her current form, before charging towards Kobold 3, jaws agape for maximum effect. She roars and, (if reaching the kobold doesn't take a dash action) swipes at the kobold in front of her.
Claws: Attack: 12 Damage: 12
If she kills kobold 3, she'll aim for K1 with her bite attack, or else she will continue mauling kobold 3.
Kobold 3 gives a gleeful laugh when it dodges Elspeth's wolf claws. It's a brief laugh. Very brief. It ends with a splatter of blood against the cavern wall. Kobold 3 is very dead.
The drake gives a roar of surprise, picks itself up, shakes, and then careens up the stairs headed towards Soar; it comes to a halt in G4 and looks intent on murdering her in the next round.
This post has potentially manipulated dice roll results.
Kobold 1 joins Kobold 3 in an unfortunate and abrupt death. Kobolds 2 and 4 shiver and squeal in surprise and alarm, but survive so far. It's now possibly dawning on them that the tide of battle has turned against them, but with no escape route, they just attack Jalt as the nearest enemy, dropping their slings and drawing nasty little daggers:
Kobold 1 attack: Attack: 24 Damage: 5
Kobold 3 attack: Attack: 18 Damage: 3
Both attacks hit; Jalt takes serious damage and is looking badly wounded.
The drake surges forward as you strike, so your sword slices into its shoulder; you feel it hit bone, and the drake shrieks, a deafening and piercing shriek that seems to fill the cavern and, you suspect, reach throughout this cave complex. You withdraw your sword, and the drake is already beginning its attack upon you.
Nate still thinks they are all doomed when Jalt takes more damage from the Kobolds. He gets a glint of hope when he sees Soar's blade enter the drake, and ignores Soar's command to fall back.
Believe me, he wants to... but he just can't leave these people that have saved his life countless times. He's not going to go alone.
He shoots a small flame from his finger towards the bottom Kobold. Produce Flame
When Nate's flame fizzles, Soar turns to him with a very, very encouraging, sympathetic smile. "You'll get them next time! I bet when you're rested, you'll be magically amazing. For now, please fall back. We couldn't bear losing you. We got this!"
(This distracted bit of encouragement, right before she's about to get eaten by a baby-dragon-thing.)
BearElspeth appears in a flurry of teeth and claws, ripping and shredding and tearing at the warrior. It's a bit of a blur, and there's no cry from the warrior. But at the end, his form - or what's left of it - lies on the cavern floor, lifeless.

The drake, provoked by the frenzy in front of it, bites at Nate while swiping its tail at BearElspeth:
Bite attack on Nate: Attack: 21 Damage: 5
Tail attack on Elspeth: Attack: 8 Damage: 9
Both attacks hit, but Nate, you feel the jaws close on your leg and the pain is intense - as the drake withdraws, you see your blood covering its mouth and a horrible forked tongue flickering out to taste it.
Jalt is up.
Follow-up question - would Soar be able to grab the warrior's leg, pick him up to swing the rest of his form at the drake with the aim of knocking the drake off into the pit?
(Sorry, was in a bit of a rush this morning - will answer about Soar and push/grapple attacks later today)
Upon seeing the warrior dispatched by Elspeth, Jalt immediately turns his attention to the drake and casts Sacred Flame 4
Hoard of the Dragon Queen - Jalt Rhogar - Dragonborn Cleric (9)
"Ive heard it both ways."
Drake's DEX save: 10
The drake takes its first injury, and turns its head to snap at Jalt.
The kobolds seem taken aback by the death of the mighty warrior who led them, but they keep their distance and attack you all again:
On Soar: Attack: 15 Damage: 3
On Nate: Attack: 5 Damage: 3
On Jalt: Attack: 14 Damage: 5
On Elspeth: Attack: 17 Damage: 5
This time, only Elspeth takes a rock to the face.
Soar and Nate are up.
(What you're describing for Soar is possible. Essentially it would be making a shove attack on the drake; using the warrior to achieve that is part flavour, but also using your environment in a creative way - I might give advantage or disadvantage depending on whether I felt it was easier to push the drake using the large dead body than just pushing. However, I would warn you that the pit isn't all that deep.)
It seems the Drake has clipped the femoral artery in Nate's leg when it took a bite, and Nate is bleeding... bad.
"Ahhhhhhh!!!! GUYS!!! IT'S OUR LAST CHANCE! PLEASE!"
Nate tries to defend himself the best he can by jabbing the Drake with his 2 daggers, basically trying to keep the beast away from him. (he doesn't know what Soar is planning yet, of course.)
2 weapon fighting
Attack: 22 Damage: 3
Attack: 21 Damage: 1
Nate, both attacks hit, but only really graze the drake. Its head is twisting back and forth between you and Jalt, unsure who it wants to eat first.
Soar is up.
Soar looks to Nate, very bloodied again, and Jalt, taking big hits but still standing, and says, "Fall back! We'll cover you!"
She looks at the warrior's body, thinking about trying to use it as a cudgel to knock the drake into the pit, but tossing aside that idea. She rushes at the drake, attempting to shove it into the pit, to buy them some time and remove the immediate threat from the board. If she can get some help from Big Brown Bear Elspeth, she will happily accept it.
Soar athletics check with help (with no help, ignore second roll): 12)
(Didn't roll with advantage... here's the second roll if Soar get advantage: 22)
Drake's Athletics check: 11

Though Soar's shove is clumsy, the drake is taken completely by surprise; it tries to swivel its tail, but hesitating between using it to keep its balance and attacking the paladin, the creature is knocked down fifteen feet onto the lower level, taking 3 damage from the fall.
Round Four
WarriorElspeth
Drake
Jalt
Kobolds 1, 2, 3 and 4
Nate and Soar
Elspeth is up.
Elspeth shoots Soar a grateful look - which doesn't quite translate to her current form, before charging towards Kobold 3, jaws agape for maximum effect. She roars and, (if reaching the kobold doesn't take a dash action) swipes at the kobold in front of her.
Claws: Attack: 12 Damage: 12
If she kills kobold 3, she'll aim for K1 with her bite attack, or else she will continue mauling kobold 3.
Attack: 17 Damage: 12
Kobold 3 gives a gleeful laugh when it dodges Elspeth's wolf claws. It's a brief laugh. Very brief. It ends with a splatter of blood against the cavern wall. Kobold 3 is very dead.
The drake gives a roar of surprise, picks itself up, shakes, and then careens up the stairs headed towards Soar; it comes to a halt in G4 and looks intent on murdering her in the next round.
Jalt is up.
Jalt will circle around Soar and move to H3 and let out a breath of cold air at the three kobolds...(con 12 save, half damage on save) 10
Hoard of the Dragon Queen - Jalt Rhogar - Dragonborn Cleric (9)
"Ive heard it both ways."
CON saves:
Kobold 1: 16
Kobold 2: 5
Kobold 4: 9
Kobold 1 joins Kobold 3 in an unfortunate and abrupt death. Kobolds 2 and 4 shiver and squeal in surprise and alarm, but survive so far. It's now possibly dawning on them that the tide of battle has turned against them, but with no escape route, they just attack Jalt as the nearest enemy, dropping their slings and drawing nasty little daggers:

Kobold 1 attack: Attack: 24 Damage: 5
Kobold 3 attack: Attack: 18 Damage: 3
Both attacks hit; Jalt takes serious damage and is looking badly wounded.
Nate and Soar are up.
Soar is... alarmed at how quickly the drake got back up. "I meant it when I said fall back! I'll hold them off!" she shouts to Nate and Jalt.
She moves down to F5 to both allow Jalt a path to the exit and also to flank the drake. She swings the longsword, aiming for its head.
Attack: 25 Damage: 14
The drake surges forward as you strike, so your sword slices into its shoulder; you feel it hit bone, and the drake shrieks, a deafening and piercing shriek that seems to fill the cavern and, you suspect, reach throughout this cave complex. You withdraw your sword, and the drake is already beginning its attack upon you.
Nate is up.
Nate still thinks they are all doomed when Jalt takes more damage from the Kobolds. He gets a glint of hope when he sees Soar's blade enter the drake, and ignores Soar's command to fall back.
Believe me, he wants to... but he just can't leave these people that have saved his life countless times. He's not going to go alone.
He shoots a small flame from his finger towards the bottom Kobold. Produce Flame
Attack: 10 Damage: 8
The little mote of flame fizzles as Nate throws it, fading from existence before it gets near to the kobolds.
Round Five
WarriorElspeth
Drake
Jalt
Kobolds 2 and 4
Nate and Soar
Elspeth is up.
When Nate's flame fizzles, Soar turns to him with a very, very encouraging, sympathetic smile. "You'll get them next time! I bet when you're rested, you'll be magically amazing. For now, please fall back. We couldn't bear losing you. We got this!"
(This distracted bit of encouragement, right before she's about to get eaten by a baby-dragon-thing.)
(Also, how injured does the drake look?)
Elspeth leaps towards Jalt's aid, maneuvering herself to J2 so she is next to the two remaining kobolds. swiping first at one, then biting the other.
Bite - Attack: 24 Damage: Unable to parse dice roll.
Claws - Attack: 15 Damage: 11