Many campgrounds have laundry facilities. When you camp in a campground, you are free to go in and use the facilities ... they usually cost a few coins to operate.

If the campground does not have laundry facilities for some reason, many towns have laundry facilities. There's nothing stopping you from taking a break in the middle of the day at a town with laundry facilities and having your lunch while doing your laundry.

I will wear the same cycling clothes for 3-5 days and then switch to my back-up set for 3-5 days. I can usually find a laundry facility of some sort in that time, and that's where my sarong comes in. I will use it as a blanket or pillow, but when it is laundry time, I change into my sarong, and wash everything else.