Would is be appropriate to use these common words that are confused in my formal paper?

Their, there, they’re
Affect, effect
Its, it’s
your, you’re
then, than
accept, excep

Or do I eliminate the ones with an apostrophe?

Don't use contractions in an assignment or formal paper. Eliminate 'they're / it's / you're / can't / didn't / won't etc.

Learn the difference between:
affect, effect
then, than
accept, except