Just bear in mind that Carlsbad Caverns and the Four Corners are on diagonal opposites in NM....and you're looking at an all day drive from one to the other.
I've made close to that trip several times as my family lived in Shiprock, which isn't too far from the Four Corners, and my wife's family was in Carlsbad. (We met in college at New Mexico State in Las Cruces in 1976.)
Alamogordo to Carlsbad is a short four hours if you go across to Ruidoso, over to Roswell and then south through Artesia to Carlsbad. The national park is about a half-hour further on.
If you do get to the Caverns, check out the lit ceilings in the lunchroom....that my wife, working for her father, helped install in the early 1970s.