I've always thought he must use a public restroom in another part of the building or in another building on campus. That doesn't explain where he would shower, though.

If I were the writers, I would've shown him in one of the sequences: at the beginning of a scene walking out of a public restroom, at the end of a scene walking into a public restroom, or have him run out of his parents' house maybe with a towel or his mom saying something like "Cory, you can't come here to use our shower all the time!" If I used that last bit, I would've fit it into the episode some how.

OK, that's just me.

I never knew where he went either.
Odd, very odd.