Bring plenty of water, and don't get stuck. You cannot depend on cell service in remote areas. Keep your tank full when you get a chance. Get out and walk in an area if you are not sure about the traction. Bring some equipment to get unstuck like a shovel, come along, tow strap, handyman jack, etc. You can be a long way from help. Use some caution. 4wd just gets you stuck in worse places.
Timing is everything when hitting the deserts. NM is cold compared to southern AZ in winter. All but the low deserts can have some cold weather. Too much heat is the worst. It is hard to run any AC without a large generator and there is little shade. Bring some. Keep an eye on your pets. Elevation makes a big difference.
There is lots of room out there for everyone. No need to camp with other people unless that is all you are used to.
I will be out near Death Valley next week and will hardly see anyone.