we always bring a scrunchie.

What's cleaner than soap? At the end of my shower, I lather up my washcloth with soap, scrub it to itself, then rinse thoroughly with hot water and hang up to dry. Isn't that almost like running it through a washing machine? In fact, the way I scrub it, it's probably better. Meanwhile if we are moving on that day, I then put the washcloth into a ziploc, and hang it to dry at the next destination.
Actually I do take several washcloths -- usually older ones, and at some point in the trip use one to scrub my shoes or filthy hiking boots and then throw it away.

I can't believe that anything that is soaked & removes dead skin from your body is going to be hygienic after a couple of days

We always use Sea sponges whether away or at home, they squeeze up real small and you dont have to worry about trying to dry them out afterwards, they dry out naturally, a few times a week I pour a kettle of boiling water over them to clean them thoroughly

take disposable washclothes.
Nothing wet to take along. Throw them away after one use.