I typically use the Chisos Basin campground because it's centrally located and is where the restaurant is. You have flush toilets and water points there. However, it's restricted to 20' or less trailers due to the road going into the basin and the small size of the camping spots. Only one loop allows generators and you may have trouble finding 5 sites open on that one loop.
My next choice is the Rio Grande Village campground. There are plenty of spaces in the generator zone. You have flush toilets and water points there. You have a convenience store nearby with gas and showers.
Even though there's about 150 miles of paved roads in the park I don't find it an issue to stay in one campground for my entire visit. Roaming the roads is pleasant (and I do it on a scooter). There are evening programs that alternate between the Chisos Basin and Rio Grande campground areas. I try to make all I can. You may also find daily Ranger-led walks that appeal to you as well.