We have camped at Cottonwood in Canyon De Chelly and it's fine. Didn't see much of any dogs, and the sites will hold 38 foot coaches. We thought it was actually quite nice and stayed several days before heading on to Moab a couple of years ago. There are even a couple of "pull through" sites.
If there is lots of water in the canyon bottom, take the time to do a 6 wheel open air bus tour. They have full day or half day tours. When we were there the canyon water was high and it was a blast.
For those who like to hike, we signed up for a "White Horse" canyon hike with a Navajo. It was about 4-5 miles round trip starting at White House trail and ending up back on the ridge. It was very informative; but that may depend on your guide.
Marsha~