As others have said, NM is a pretty spread out state and you should consider a regional visit as an option. Like Second Chance has said finding any kind of RV spot down in the Southeast corner can be difficult due to the oilfield and pipeline activity. I commute through Carlsbad frequently and most of the RV "parks" have No Vacancy signs out. Even the bare gravel lots that they call parks.
However, if you do make it to Carlsbad and the caverns please save some time to drive another 45min Southwest to Guadalupe Mountains NP and see the highest point in TX. Spend some time at the visitor center and try to visit the Frijole Ranch site to see how folks lived when the area was being settled.
Let me know if you have specific questions about the area, I am living in my trailer here now.
Terry