I shower in the campground or hostel shower facilities.

Or if for some reason those are not available:

-- I swim in lakes, rivers, or the ocean
-- I shower in the beach showers
-- I shower in the truckstop showers
-- I use sinks in toilet facilities to take "bird baths"
-- I use baby wipes.

As for keeping clothes clean ... the dirty clothes go on the top of the clothing pannier, the clean clothes are on the bottom. When they are all dirty, I locate a laundromat or use the laundry facilities in the campground or at the hostel.