An RV has a lot of advantages over trying to stay in motels. That is especially true if you travel a lot and want to see more remote areas. It seems many people get talked into or otherwise buy large RVs. They are going to be expensive, require maintenance, are way more difficult to drive and park and will greatly restrict where you can stay.
Forget the laundry. Instead think about a much smaller unit. What do you really want or need? For me it is a comfortable Queen sized bed, a small refrigerator/freezer, a 3 burner stove, a sink, and definitely a wet bath with a shower. A medium sized truck camper worked fine when my wife and I were full timers with a couple of cats.
Laundry is easy and an RV park is usually the last place I would go. Every major town will have a laundromat. We needed to stop about once a month and filled up a bunch of machines at once. Even living at home I have grown used to the advantages of having a month's worth of underwear, socks, and pocket T shirts. They do not cost a lot or take up much space or require me to do laundry more often than every few weeks.