Is there something in Moab your specifically going for? Otherwise I'd personally consider heading further south or west and avoid the Moab crowds. There are not many secrets left in the Moab area for camping. I think the suggestion of Behind the Rocks is probably the best. If you strike out just goto Area BFE.
Lots of hiking down in Comb Wash, like Arch Canyon and far less people. Indian Creek? Or out in the Swell to the west. I love Moab like everyone else, and that's the problem now. No more pulling into Sand Flats Friday night and picking out your favorite spot.
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1976 FJ40, Project with little progress.
2004 Starcraft 10RT