You will find everything you are seeking at one of the campgrounds near Big Lake. Reservations are a must, especially this year. Sounds like you are fine without hookups and if so try the Rainbow CG which has very large sites, flush toilets and water spigots throughout the very large campground {and no you can not fill your RV from the spigots}. Showers are available but it is a bit of a hike and cell service is next to nonexistent. There is a dump station just outside of the CG.
There is another CG just down the road with hookups, I think the name is "Trout" check the Arizona Campground Map {Google it} for details and the reservation number. Both campgrounds are located at 9,000+'. We were at Rainbow a couple of weeks ago and there was a complete fire ban in effect. They have had some rain since then so it may have been lifted. Expect serious thunderstorms most afternoons but then that is part of the charm of the high mountains.
:C
