So many great things to see and do in both states. Make sure you the caverns in both states. Take the tour in Carlsbad, certain tours they take you deep inside and have you sit on a bench. Then they turn the lights out!
Do make a list of the tourist things to do in both states if this is your first trip there. I have also camped far off the highways many times in both states. I rarely had cell coverage. I'd recommend one of those emergency sat devices that several companies sell if your worried. Well worth the peace of mind if you don't have ham radio as a backup.
Definitely keep a firearm and flashlight in your lap when out enjoying the night sky. I have been stalked far to close for comfort by various critters and a good bright flashlight usually scares them away. But I still keep the handgun ready just in case one or a group are just to hungry to pass up some "people buffet" food!