There are a lot of RV parks in Moab area, we stay at the Canyonlands RV Park right in town because we can walk to shopping and restaurants. For dry camping near Moab, we like Horsethief Campground on the road into Canyonlands National Park (hwy 313), it is a BLM campground and usually not crowded. Near Monument Valley we have camped in Valley of the Gods, it is a 17 mile good gravel road off hwy 163 near Mexican Hat. There are no specified campsites or facilities, you just find a good spot off the road and enjoy the spectacular views.
Have a great trip,
Bob