Warning - Biased opinion
In a little over two months while volunteering at Davis Mountains State Park, I looked at all the other RV 'parks' in Fort Davis. I wouldn't waste my money on any of them. I said it was a biased opinion.
If I couldn't get into Davis Mountains SP, I'd go to Lost Alaskan RV in Alpine. I would spend my money to stay there.
Carlsbad - as an Escapee - I stay at The Ranch, but Brantley Lake State Park would be fine. Dog Canyon is on my to do list, but I wouldn't use it as a base to see the caverns.
I haven't been impressed with any of the RV parks in Carlsbad, but the KOA near the state park isn't too bad, just expensive.
It really comes down to how much driving you want to do.
White's City RV Park is overpriced and in the middle of one of the few "Great American Tourist Traps" left in this country. But you cannot beat the location for seeing the caverns.