Grammatical function
Solution 1:
while we sat holding our breath.
while we sat holding our breath is a free modifier. It modifies the main clause as a whole.
A free modifier differs from a bound modifier in that a bound modifier attaches to only one word or phrase.
From "ThoughtCo."
Definition:
Generally, a free modifier is a phrase or clause that modifies either the main clause or another free modifier. Phrases and clauses that can function as free modifiers include adverb phrases, adverbial clauses, participial phrases, absolute phrases, and resumptive modifiers.