I have a camp towel that is quick-drying (an MSR camping towel). It's more like a chamois for your body/hair. It dries quickly. I can hang it somewhere in my office and it dries in an hour or two.

Maybe you could just cart a small toiletry kit with you everyday. Something with travel sized things that wouldn't be big/bulky to carry with you everyday since your shower situation isn't very convenient. The toiletry kit I carry with me when bike touring isn't much smaller than the one I use at work. It has a folding brush, travel size toothbrush/toothpaste, shampoo bar that I use as shampoo and soap and a small container of deodorant (I make my own deodorant, so I put it in a tiny screw top container).