We have done laundry while on the road and yes we ran the generator.

Doing laundry while driving seems counter-intuitive but if you are going from campground to campground it really doesn't matter if you use a lot of water because you will be refilling that evening. It is a good use of time to do laundry while driving instead of using time when you could be out and about.

Contrary to what Jim says my wife avoids using public laundry machines because she is unsure what has been in them before she uses them.