Been awaiting more replies to this. Like others have said, we've always stayed at either Guadalupe Mountains NP,(in TX) or Bottomless Pit SP(in NM). Both are really nice.
Guadalupe mtns, wherein resides Texas' tallest mountain is in the middle of nowhere, and has great hiking.
Bottomless lakes is close to Roswell, but doesn't feel like you're next to a big town. Others have posted their pics somewhere on here, but I couldn't find them. The camping spots along the "big" lake are real pretty. Beautiful sunsets. The salt-water is nice and refreshing to swim in on a hot day.
Back to the original question...isn't there any place closer to White's City or Carlsbad, NM?
Can't vouch, but check this out...
http://www.whitescityrvparks.com/
I'll give it a try next time we head out that way.