We wouldn't utter a single word, (we would) just listen to each other's feelings echoed in our songs.
Can I omit the we would in the sentence above? Why or why not?
Yes you can, with a few modifications. Reason is because both clauses share the same subject and verb form, thus one clause can be omitted.
The modification needed is a conjunction and removal of the comma. Since the first clause is positive and the second is negative, the proper word is but:
We wouldn't utter a single word but just listen to each other's feelings echoed in our songs.