Which form is correct?
His palms were sweaty and fingers trembling, but...
or
His palms were sweaty and fingers were trembling, but...
Which form is correct?
His palms were sweaty and fingers trembling, but...
or
His palms were sweaty and fingers were trembling, but...
They're both correct.
Let's start with an even more clearly correct equivalent:
His palms were sweaty and his fingers were trembling, but…
Here we have two simple clauses, joined with and. There's nothing one could complain about, or anything puzzling.
Your two examples each elide either the pronoun that is part of the subject (his) or the pronoun and also the verb (were) from the second clause.
The main thing here is how obvious the elision is. You have to elide quite a bit before something becomes ungrammatical, but it can become bad before then (having writing that is grammatical and horrible to read is worse than having writing that is technically incorrect by some rules, but clear and enjoyable to read).
Both are obvious enough to be understandable. We would probably want less elision in a formal context (something closer to my example above than either of yours), but from the meaning this sounds like a case where the benefit to the pace is more important than anything else.
You might also consider the different elision:
His palm were sweaty and his fingers trembling, but…
I like leaving in the his because while I'm leaving out the were to hasten the pace and convey his excitement, keeping in the repetition of his could help keep a focus on "him" and his emotional state.
That said, I might disagree if I saw the fuller sentence after the but.
All four (the two examples in your question and the two in this answer) serve the same task technically, and the elision in all of those which elide is justifiable, so you're left picking between them on the grounds of which to your ear best serves the emotions you want to convey or simply sounds better; which is a matter of taste.