If you are looking for places to disperse camp in and around Moab down to Monticello, send me a PM. I just moved up here from Moab, after living there for 6 years, and believe me, I know of all the BLM campgrounds and places to disperse camp.
BLM has no limits on the number of dogs in their campgrounds, just a regulation that they have to be on leashes; however, the Rangers usually accept staked dogs as long as the owner is out with them. Most important in the campground is they don't bark and disturb other campers.
But that entire area from Monticello north to Moab area has thousands of placed you can disperse camp and not have a neighbor; depending on when you will be there.