Thanks to my friend Ellen's comments, I used the USGS website (not actually that easy) and found what I think is a good candidate: The Twin Forks MDWCA. It's outside Alamogordo (which is why I would have thought Albuquerque), it's directly east of Tucson, and I remember we passed through Roswell the next day, marveling at all the alien iconography. The only problem is, this isn't a city; it seems to be a reservoir, in Mayhill New Mexico, which is part of New Mexico's Mutual Domestic Water Consumer's Association, which (in turn) is apparently responsible for providing safe water to New Mexico's rural areas. So if I had to guess, maybe Mayhill grew in the last 15-18 years and swallowed up Twin Forks. At least now I have an idea of which Chamber of Commerce to call.
LATER: Damn. It turns out that on this computer I'm borrowing, I can't seem to add links in Blogger. I dont know why, but I have writing to do and don't have time to futz around with stuff like this. So if you're curious, just go to Mapquest or some such and enter "Twin Forks MDWCA, NM." You may have to pull back a few steps until you can see it, Alamogordo, Roswell, and Tucson all in one screen, but it's worth it just for the closure.