As far as I have heard, Aguirre Spring Campground is still open. It is a BLM site, fairly high up, should be cool at night. Nice views. Good price for a stay. Not to far from El Paso. It is located on the east side of Organ Mountains, east of Las Cruces. From Cruces take U.S. 70 for 14 miles east of the I-25/U.S. 70 interchange in Las Cruces, then turn south on the Aguirre Spring Road (1.1 miles after San Augustine Pass) for six miles. At mile 4 the road becomes a one-way loop and climbs steeply. There is a gate at the start of the one-way this entrance gate is open from 8:00 a.m. until 7:00 p.m. (Note: you can leave but not enter until after 8 a.m. or after 7 p.m. in case you are not back from the concert late at night if staying there).
Also another BLM site: Valley of Fires Recreation Area should be open. Four miles west of the Town of Carrizozo on US 380. Carri'zo is north of Alamogordo via US 54.
Lee Memorial State Park is 12 miles south of Alamogordo via US 54, then east 4 miles on Dog Canyon Road. Us 54 also runs down to El Paso. The park will be open but the Dog Canyon Recreational Trail is closed indefinitely -- the Lincoln National National Forest is closed due to extreme fire danger. Nice showers, electric hook-ups if you need them and water as well at the state parks.
Along I-25 Leasburg Dam State Park will be open. It is 15 miles north of Las Cruces on I-25, get off at exit 19 and take NM 157 west to the park entrance. You may fish on the river there.
Elephant Butte Lake State Park will be open also five miles north of Truth or Consequences via I-25 exit 83. Fishing on lake shore or by boat, also nice fishing on Rio as it leaves the dam (Trout with barbless hooks only). Some camp sites along the Rio as well.
You can also do dispersed camping by Corona (US 54) in Cibola Nat'l Forest, one area close to the hwy is a mile or so south of Corona, turn west @ CR A027 <I believe it is> juniper/piñon. Farther south of Corona by several miles (also west turn) off onto CR A019 will take you back into Cibola to higher country into Ponderosa trees (but several miles on the dirt road). The Gallinas Mountains are a small isolated range west of Corona, New Mexico. The elevation at Gallinas Peak, the highest point, is 8,637 feet. For people traveling from the east, it is the first major mountain range they encounter along U.S. Highway 60 and 54. The Gallinas Mountains support a small elk herd of approximately 30 animals. Mule deer, turkey and black bear are also present in addition to rabbits, bandtailed pigeons, badgers, bobcats, and porcupines. At this time: The Gallinas Mountains near Corona are open.
I do not remember if you have A/C or not. If not some of these maybe a little warm during the day, but should cool down at night.