We also stayed at the KOA north of Carlsbad (the town). Nice park, but about 30 miles or so from the caverns. Been a few years, but no other RV park in the area had decent reviews. Check rvparkreviews.com. The RV park at White City is convenient but had very poor reviews. I guess it would be ok just the night.
Back in 2014, we stayed at the KOA just north of Carlsbad. There was a bit of a drive to the Caverns which are a little south of Carlsbad. Reviews indicated the KOA was much better place to stay than the campground at White City.
That may not be the case now. A lot can change in 5 years.
Erroll, Mary 2021 Coachmen Freedom Express 20SE 2014 F150 Supercab 4x4 w/ 8' box, Ecoboost & HD Pkg Equal-i-zer Hitch