Well you cannot visit all of these places from one city. Carlsbad Caverns is over a 3 hour drive from the dunes in White Sands National Monument. The hotels down there have gotten quite expensive due to an influx of contractors from the oil industry. You might do the drive from El Paso (early start!), visit the caverns, and then start heading toward White Sands, staying somewhere along the way like Artesia (although accommodations can still be pricey there). The next day, go on to White Sands and then stay somewhere around Alamagordo, or north of there if you think you can keep going. From there, you can get to Albuquerque the next day and Santa Fe is only an hour north of Albuquerque.
You never mentioned how much time you had for this trip. A lot of people don't realize that New Mexico is the 6th largest state and traveling from one area to another takes much more time than most think. Best of luck.
You really should give the caverns a full day; main entrance, big room and a guided tour of the Kings Palace, is the most recommended use of time.
You could stay in White City, just outside the Caverns. The Hotel there is o.k., nothing fancy, but a lot less expensive than in town. Artesia is somewhat less expensive,
White City does have a small restaurant, don't know the quality or the open times on it. That is why I recommend Artesia as there are several good restaurants there. It is a longer drive to get down to the Caverns but an early start will give you most of the day.
As far as Albuquerque and Santa Fe, I would rethink those. It is several hours drive to Albuquerque or Santa Fe from either White Sands or Carlsbad. It really takes a good part of a day, so unless you have over a week planned, I would rethink those places.
The replies here make me wonder: how reliable are the travel estimates if I use Google maps to give me drive times from one NM location to another?
Pretty much rule of thumb, take the Google time and add 30%, and you will be close to actual time. Google doesn't figure in stops for restrooms, food or gas.
Hope this helps your decision.