I think a rental is your best option if you really want to use an RV, but looking online, you are probably going to spend at about $80/day plus mileage and gas. All told, you're the $3000+ neighborhood for a month on the road. Add in campground fees and I think you'd be better off traveling via passenger car/van and staying in hotels.
If you decide to go by RV anyway, you can get a 10% discount on Cruise America if you have a KOA card. The card is inexpensive, and you save 10% at KOA parks as well.
I've been to all those places, and they are definitely worth seeing, but rough time of year to go. It can't be helped, so have fun!
Elizabeth