We just got back from Moab also, spend almost 10 days there. We stayed at Portal for a week and then Dead Horse Point State Park for a few nights. Portal is very nice and is just outside of town on the north side, so it's close to Arches Natl Park. Socializing with other campers isn't a factor for me, so no issues there.
We also stayed at Dead Horse for a few nights, and it's a pretty awesome place to camp but it isn't convenient to much else. But if you're only staying for a night, it may not matter.