I've never found any issue with dogs in the Moab area. Including SRCG. This year I'm at Portal which is directly behind SRCG. But is still a private campground. Just much cleaner.
Any of the BLM campgrounds along the river should suit your needs if you can find a spot. Over the years many of the primitive camping spots have been shut down around the Moab area. Everywhere is over run IMO. Has trash all over, vandalism, etc... Moab has started to bring the suck that way.
Speski OffRoad LTD, Cargo Solutions for Toyota 4x4's http://www.SpeskiOffRoad.com
"80 series, with a fridge"
1976 FJ40, Project with little progress.
2004 Starcraft 10RT