On the one multi day tour that i’ve done I followed the one to wear one to wash advice. The only additional thing I did was take a little mesh wash bag with me so that i could put the shorts inside and attach them to the top of my panniers so that the air could get to them during the dry weather without them flapping a round like some sort of weird flag. I know you’ve got no panniers but I’m sure you could find a place to attach them if needed.
Washing really wasn’t a problem and drying was just a case of being a bit inventive dependent upon where you were sleeping that night and the likely weather. Sometimes inside the tent, sometimes out and on the luxury nights they’d be strung across the bathroom of the B&B. A good wring out and exposing the chamois to the air would see them dry pretty quickly.
I’d second the sudocrem suggestion. They do a very small, perfect for travel sized tub.