So, the dermatology team came in to look at his skin.
They determined that this was not a reaction to the antibiotics, but was instead a skin peeling condition called keratolysis exfoliativa which can happen to people with sweaty hands, but is apparently very common with CF. Basically they said now that he has this condition, he will always have it, but they can prescribe a lotion that can help the dead skin come off faster.
They changed one of his antibiotics, Gentamicin to Tobramycin. They will continue to watch out for any hearing loss from the medication.