RV Park at White's City. That is where I stayed, quite close to the caverns. Can even hike to the caverns from there if you have the time. 3.7 miles one way. But I went there on my electric motorcycle on the paved road, but also did some hiking. Besides, I did the hike in and out of the caverns, didn't use the elevator.
They have both boondock as well as full hookups. By far, most at the full hookups. I was alone out in their boondock area, no need to use a headphone with my ham radio stuff out there during the night. I was far from all the other RVs. Plenty of room there, I don't think that place is ever full, and the boondock area there had NOBODY except for me.
And at least there, they said I can dump my tanks for free at any of the unused full hookup sites on the way out.
Nice restaurant right across the street, too.
I see the RV park has a lot of negative reviews, but I had no complaints at all.
That was three or so years ago, on my way back from the Big Bend of Texas. The worse paved road I ever been on. The countless large potholes are impossible to miss. I believe that is what bent up my stirring rods and made my wheel alignment impossible. But I made it back okay.
Just stay out of that part of Texas (unless they repaired the road). The same road is fine when once in NM.
-Don- Reno, NV