I personally think that they’re usable in certain circumstances like instances where it would sound weird without them, but some people have been telling me to delete them. But if I do that, then I would have to delete some words as I’m at about 647 words. Thoughts?
I’d leave them; this is formal writing but not stuffy. You want your voice to come through.
^ Agree. You do not use them in formal writing, but this is one step below pure formal. Use them with caution.
It depends on the actual contraction.
I think “don’t” is normally fine, since “do not” tends to sound too stilted.
But “could’ve” is just too informal.
But still a step up from coulda. 
I agree with the others; there is no need to avoid them, you do not want it to appear too stilted.
And, they save you from bypassing the word limit 