Whitney lake is nice, goto Evanston then back into the Uintahs. You may want to plan on arriving Mon - Wed. It may be hard to find a spot on the weekend.
If you want to drive a little, the area around Goblin Valley would be excellent for camping + RVs. Go to goblin valley, but instead of going into the park stay on wild horse road, there are numerous unimproved places to camp around there, and RV roads. Plus little wild horse is an easy slot canyon but still beautiful if you want to hike.
I will talk to my bro in law, he has RV's and camps the way that you like to, I am sure that he will have more spots.