Can't offer much advice. I've been to north east NM a couple of times, but didn't look carefully at the trees. Recall a fair number of cottonwoods near river courses. That wood has rather long fibers, and is quite soft. Its OK for carving, but I have doubts about is worth for stickmaking. I recall a lot of pines in the mountains. Pine can be strong, but again, the fibers are long, making clean cuts more difficult, and there are often resin pockets and resinous knots that aren't desirable. Have vague recollections of white barked trees, aspen I suppose, or birch. I've seen some nice work in aspen, and birch should be good for sticks.
As to elm, most of the American Elm where I live died off decades ago from Dutch Elm Disease. Supposedly it was quite good for furniture. There's some slippery elm around here. From what I understand, its never been used much. Also lots of imported Chinese elm. Being a very fast growing tree, with a tendency to bleed sap, I doubt it is worth trying.
I've seen a few nice handles made from mesquite, which I suppose grows in NM. The wood is very hard, from what I've read, but finishes nicely.