Did that trip a few years ago. We took the route thru Alamorgordo, stopped at White Sands and ended up at the very nice KOA just north of the cavern. The two high points of the cavern was the lantern tour we took and sticking around tip dusk and seeing the bats leaving, quite a sight!
We were in a small MH and I don't recall how the road was, but I do recall driving thru an unbelievable thunderstorm.
After Carlsbad, we headed up to Roswell and the UFO museum. Spent 1 hour going thru it. The DW said it was 59 minutes too long.
Enjoy the trip!