Full size rigs are permitted inside park boundaries on designated trails. The ATV rule seems to be NP policy, probably due to riders failing to tread lightly, affecting the ecosystem.
I recommend the National Geographic Trails Illustrated set of maps. There's one just of the Moab area showing mtb bike and 4wd trails - green for easy, blue for medium, black for hard. There's also Mountain Bike America (Moab) by Lee Bridgers that contains a lot of trail descriptions and elevation profiles that I find quite detailed.
Outside of NP areas, I suggest Klondike Bluffs, Gemini Bridges, Onion Creek, as easy rides.