So I am in a campaign right now where we have a full cleric in the group as well as two of us that have taken a level of cleric multi-class.  I'm playing a Drakewarden with a level of War Cleric for extra bow shots and utility. My companion is a War wizard with a level of Forge cleric for AC.  I find I have no extra spell slots by the time we rest due to using Pass Without Trace or Goodberry along with bringing my drake back from death.  What I seem to use my slots for the most however is Healing Word.  

The wizard seems to horde his spells for battle attacks only.  We tried to buy him a scroll of Find Familiar but he stated he hates pets.  When he spoke of not having good bonus actions we suggested swapping Cure Wounds for Healing Word but that was a nonstarter.

We were fighting a boss last week where they had a damage aura.  Because my drake essence matched the aura she was unharmed but the barbarian went down three times and got hit with the yo-yo word three times from the cleric or myself. Now the cleric is Order so that grants an attack to the Barb so worth it but I felt like my slots were being drained. 

Is the wizard correct in saying keep yourself alive and just saving slots for big hits?  Should I be more stingy with my slots and risk my companions taking two death saves instead?  Am I upsetting the game balance by keeping the team from needing to roll up new characters more often?  Am I an arse for suggesting tactics like using cover?

Both are valid ways to play. 

A big thing is to make sure you are using your spell slots judiciously.  Don't top up people in combat just because you have the bonus action available...remember you can't out heal the damage output of most challenge rating appropriate monsters, so in combat healing word really should be mostly about emergency defibrillation of people that have actually dropped.  Because of this Healing Word is almost always a better choice than Cure Wounds if you have to pick one or the other...you get to do it at range and there isn't much difference between 1d4+wis or 1d8+wis when the dragon is going to bite you for 2d10+8 or breath fire on you for 18d6.

Also, out of combat make the players use their HitDice for short rest healing before you break out your spells.  A lot of adventuring days go by without having used all of your hit dice...and you don't get bonus points for having resources left on the table like that.  Cure wounds is a better out of combat healing spell than healing word is...but the best out of combat healing is hitdice.  The barbarian's 1d12+con is better than your 1d8+wis...and they probably have more hitdice than you have spell slots.

I would recommend taking a look at the Dungeon Dudes video "How to Play a Healer".  They cover these things and a little bit more.

It is better to avoid taking the damage then giving the healing so tactics like taking cover is important, but so to is preventing the enemies being able to harm you through the use of spells (either through something like hold person or by reducing hit points to 0). 

I would never leave a PC unconcious you are going to use a spell to get them up anyway and if you want the unconcious person up so they do not waste their action (and bonus action and possibly a reaction). On top of that if the bad guys have any intelligence and recognize (at least) 3 of the adventuring party can magically heal their unconcious ally then they will make sure to kill them (Many DMs wouldn't do that because they don't want to kill off the PCs but if I was DMing and the players were doing that I would warn them it is a possibility and if they have revivify might just do it)

I would also agrree with bounces that if you do get hurt it is nearly always best to use hit dice howeverf I would point out that you only get half your hit dice back each long rest (rounded down). A campaign I am player in is currently level 3 and have 4 days to get as much loot as we can from a dungeon, with us only getting 1 hit dice back each night does mean you don't want to use all your hit dice at the end of the adventuring day.

Something else that just occurred to me is that it shouldn't be surprising that the Ranger is running out of spell slots...even with a cleric dip, your main class is a half-caster.  If you find yourself desiring more magical umph you might want to invest even more in cleric.

Healing word should really be used only when someone goes down to 0 hp.   Most other healing is best used out of combat, with few exceptions.

You know what keeps your Party from going unconscious? Dead enemies. Controlled enemies. Held enemies. If they can’t attack or are limited in their options the less you need to spend your time healing.

One thing you can do is Focus fire on one enemy at a time. Two goblins with 1 hp can still attack and do 5hp damage each for a total of 10 damage per round. But if one is dead and the other is at full health because you focused on taking down one at a time then you cut the damage they are doing in half. 

Not saying you are doing anything wrong, we don’t have enough information, but maybe the wizard has the right idea and is trying to keep the rest from taking damage by killing the enemies. As others have said, healing in combat is a losing strategy unless it’s to get someone back up from zero. Enemies just do too much damage. 

And control, buffs, and debuffs can be just as important as dealing damage for the survivability of your group.