I was wondering about the effects of Burning Hands and Shocking Grasp spells while holding weapons like daggers and shortswords. Do the spells get added to the weapon damage or are they unarmed damage?
What Filcat said. To clarify, since both spells require an action to use, in most cases you won't be able to attack with a weapon in the same turn you cast the spell. And if you have weapons in both hands (or shield, etc.) you won't be able to cast them at all.
Remember there is often a discrepancy between how a spell has been described since the start of DnD in edition after edition.
As per spell requirements, both spells require Verbal and Somatic. This means the caster must be able to speak and have 1 hand free to cast.
In this case if both hands are holding something like a weapon and/or a shield you can't cast the spell. Remember Arcane Foci can only replace the Somatic component if there is also a Material Component to replace as well.
If the Caster has the Warcaster Feat, then they can ignore the Somatic requirements, but not Material Requirements, and so could cast either spell with boths hands holding weapons and/or shields.
You hold the staff in one hand and cast the spell with the other hand. I don't think you can twin burning hands. You can twin shocking grasp and since the range is touch, both targets need to be adjacent to you.