I've been to El Paso but never got to ride the trails. That state park on the west side of the mountain looks like a lot of fun.
I'd go with plain old knobbies for around there, because there's so much sand. So something with large, widely spaced knobs. There are lots of good brands around. With all the cactus, I'd also run Slime in the tubes, maybe a kevlar liner in the tire, maybe even a tire liner (ask at the bike shop) because flats suck.
Oh, you probably know this but on a steep hill your tire may slip no matter what kind of tread, because of weight distribution. If you stand up to pedal, weight shifts to the front so the back slips. If you sit on the saddle but are too upright, the front wheel is light and loses traction (control).
What I do is stay seated, bend my elbows and sort of pull on bars in a down and toward me kind of pull. This keeps the weight balanced and you can still generate power.