The Rio Grande Village has the best campground. No hookups but water points and flush toilets. (You can get hookups at the nearby General Store, basically parking lot style. Fairly decent birding with a nature walk area.
The Cottonwood Campground has water points and pit toilets. Also fairly decent birding.
Your trailer is too long to go up to the Chisos Basin campground but you should go up there to do a little birding on the short loop trail and have a sunset dinner at the Lodge restaurant.
With your rig I'd camp at Rio Grande. For one day I'd sightsee west to Santa Elena canyon, check out the scenic sites and spend some time birding at the Cottonwood campground. For a second day I'd run up into the Chisos mountains, do a short trail and have dinner at the lodge. Then you're free to roam the rest of the park and sitesee in the Rio Grande Village area. Be sure and check out the hot springs near Tornillo Creek. Old general store, old motel and hot spring still flowing. Pictographs on the rock walls just after the motel. Lots to discover in this park!
With passport you can cross the river into the Mexican town of Boquillas (this is just east of the Rio Grande Village area)