Canyon De Chelly is an awesome park though not your typical park. Unsafe, never felt that way but I would make sure my vehicle is locked. Nothing like seeing the local school surrounded by and 8 foot chain link fence topped with barb wire. It is a very poor area. The campground should not be an issue. There is another campground way up the southern side called Spider Rock. It is I would call very rustic but it is a great location. No the spots are not large, no there aren't hookups, the road within is tight in areas but at least when we have been there few if any people and off the beaten path quite secluded. It is actually quite dumpy looking when you pull in but I loved the place. must be the seclusion I liked. Have stayed there a few times. I suspect most people wouldn't care for it.
There are a variety of tour operators that run the Shake And Bake Tours with seats in the back of a truck or a deuce and half. You can also go in the canyon if you hire a local guide. Just make sure your vehicle can handle the sand and that you have room for the guide as he rides in your vehicle. IF you like horseback rides there are a couple vendors that do that. I have no idea about the short ones but we have done it a couple times in the fall when the leaves are changing and cooler, it is an all day deal. They trailer the horses to Bat Canyon you ride cross country for a bit then down into Bat Canyon which leads into the main canyon (the decent into Bat Canyon is not for the faint of heart as it is a somewhat steep trail). Once in the main canyon you go past Spider Rock, White House Ruins etc and on down canyon to the mouth. 6 or 7 hour ride The ride is spectacular imho. You take your own water and lunch.