I recently moved to NM, and prior to that I traveled here quite a bit to visit family. I called the NM State Police prior to my first visit (as I do anytime I travel to a new state) and asked the gun questions.

My TX CHL is recognized.
OC is totally legal, without any permit, and it is common in some parts of the state.
Same in a vehicle. You can leave your pistol on the dash if you like. Or keep it concealed. Your choice.
You can only carry one gun concealed. No bugs or New York reloads. I believe one concealed and the rest open is ok, but don't quote me.

I don't know about National Parks, and I haven't visited the areas you're planning to go to.

Guadalupe appears to be in TX, and that is Concealed ONLY. NO Open Carry in TX. Handguns must be concealed in the vehicle and on your person.