I've been to most of the national parks here in Utah, and I don't think I've ever camped at a designated campground. Most of the places I've been have areas you could camp at for free pretty close by (I don't know if this is the case anymore though) If you are serious about staying overnight in Utah somewhere, I'd be more than happy to ask my dad where a good place to camp is. He is a camping guru and knows all of the good, out of the way spots.
Dinosaur National Monument is pretty cool if you manage to make it up there...but like the Jazz...I haven't been in a long while, it might be neater when you are younger
