I don't know how close you you want to be to Moab, but when I was there 11 years ago, we camped at Mason's Draw Campground in the La Sals. Google Maps it and it will take you there. It is at the base of Gold Knob mountain. I bushwhacked up the mountain from the campground when I was there. Memorial Day so Mt. Waas was too snowed in but that would be a great mountain to hike up too.
Beautiful drive and beautiful scenery on the loop road that takes you by the La Sal Mountains and Fisher's Spire. A must drive if you go there. I saw some other areas around there that looked like people were car camping in non-designated campgrounds as well. There some other designated campgrounds too. If it's Labor Day, then it will be crowded. I would talk to the forest service before you go as well. The campground was not crowded when we were there over Memorial Day because the forest service was telling people there was too much snow. We drove up there and the campground was uncrowded and no snow. It was perfect. Labor Day may be a different story.
Get GAIA maps. It will tell you where you can camp. You can get an awesome discount in the show more details section of
this YouTube video. Then you can watch
this YouTube video to learn the basics of how to use it. It basically is a GPS plus a whole lot more. Records distance, altitude, etc...
Canyonlands is cool, drive all the way to the end where you get the overlook of the Colorado and Green rivers intersecting. I thought Arches was overrated and crowded, but it's probably something you may want to do once in your life. Shafer Trail is a cool road that they will tell you it is a 4x4 road. I did it in a 2-wheel drive Ford Escape. Awesome road.