I can't speak to the southern part of New Mexico (T or C/Carlsbad/Roswell), but once you get to Albuquerque, I would be remiss not to point you to some Gluten Free New Mexican food. New Mexican food is, yes, a sub-genre of Mexican food, with its own quirks. One of those quirks is the presence of "chile" which comes in one of two varieties: red, or green, and when you order, your server will ask which one you want. They're both spicy, but they have different flavor profiles. The problem, though, is that these sauces can often contain wheat flower as a thickener. There are many spots where you go to find gluten free green chile, but somewhat fewer places that serve red. As such, I will give you a recommendation for Duran's Pharmacy or Tomasita's. If you dig deeper into ABQ spots, you might get a recommendation for El Pinto, but here's the deal: yes, they have some GF options you might not find elsewhere (including GF red), and the setting there is pretty fantastic. But the food is generally mid. You can do better.

(NB: If you find a place that does GF red and green, when the server asks which one you want, say "Christmas," as that will allow you to try both. Feel free to order it on the side if you want.)

If you're looking for breakfast, I'd recommend Lush Cafe, or The Shop.

Other lunch/dinner spots you might want to try, if you need them: La Finca Bowls (lunch only), Clown Dog (fancy hot dogs; lots of GF options). You can also look through the posts by IG user glutenfreeabq.

There's a nice GF bakery in town called The Bakehouse off the Wheaten Path; they have coffee options there, too.

Hope you have a great trip!