For a while, when I would go jogging as part of my lunchtime breaks at work, I took showers at the gym there afterwards to wash off the sweat. That must have been around 20 years ago. I can barely remember how I dealt with the moist towel... I must have put it in my bag and hung it to dry in the evening after getting home. I don't remember bringing a washcloth so I probably didn't wash my butt or lower regions during those showers other than letting the water run over it; I would have done the full wash during my normal morning routine at home.