Maybe stay at Lajitas and a day trip to the park.
Lost Alaskan at Alpine is a good base. If you can get reservations for Carlsbad, I understand it is not crowded.
Yes, there will be fewer travelers, but they will overwhelm the available spots at the National Park, state parks at Fort Davis and Balmorhea. Might check on Monahans SP. I found the dunes interesting.
My experience with NM state parks this past summer was unpleasant, them being closed suddenly, and the reservation fee that they kept for each reservation. Not sure staying in NM would be a good idea.