We were just in that area. Near Moab, I would definitely visit Arches. The campground there is nice and very close to all the hikes but reservations are hard to get. You can stay at an RV park in Moab of course but we noticed there is dispersed BLM camping at Cooper Ridge about 15 minutes north of Arches. There is also primitivie BLM camprground sites at Horse Thief about 25 min away.
Another stop would be Dead Horse Point State Park. And from there, Canyonlands NP Island in the Sky District. They have 12 site walk-in campground.
Be aware, that after the turnoff for Dead Horse SP and Canyonlands, there is no where to fill up your trailer with water. You have to get water in Moab or elsewhere.
Other places to venture out to: Capitol Reef, Escalante, Goblin Valley.
You can see my recent post mostly about Canyonlands (including Needles District further south), including photos here