Yes, TUC establishes there are bunkrooms. And the Voyager
episode "Flashback" runs with this, showing a bunkroom aboard the Excelsior
which downright looks like a barracks.
I understand you're only using TOS to the exclusion of all other references, but I think that it would make sense to have bunk rooms for short-term enlisted crew. After all, it seems like the Enterprise
is always exchanging crew. Perhaps some of the lower ranks get rotated fairly frequently, so they only carry about a dufflebag's worth of personal effects and expect to share a bunkroom with x-number of fellow crew? If they're only aboard for six weeks then this seems perfectly easy. I'm not suggesting hotbunking, but several crewman to a bunk room is hardly without precedent.