I concur with what others have said regarding the area. I live nearby and raveled thru Carlsbad to El Paso at least 2x / month. The park at Whites City would likely be OK for a night. They do have a section for the long term folks that is separate from the short term portion. The other recommendation would be the KOA or Escapees park North of town.
Another option to consider would be the campground at Guadalupe Mtns. There are no hookups and it is limited both in size and number of spaces but would be a quiet option for one night. If you have never been there a stop at the visito center would be recommended since you may be driving right past there anyway. Another spot at the park I would recommend visiting is the Frijole Ranch house - it is an amazing display of how the early settlers lived in this area.
Last thing, the traffic I the area is really bad so be patient and watchful as you pass through. Let me know if you have any more questions.
Terry