Definitely no guarantee of snow in town. It could be bone dry or could be buried. The ski area just outside of town will at minimum having tubing and bunny slope running with artificial snow.

If you are staying in Alamogordo, the drive from Alamogordo to Cloudcroft won't be fun if it happens to be snowy. Also, Alamogordo itself doesn't have a lot going on. It's convenient for White Sands access and the space history museum in town is neat (but small). Other than that, it is a flat desert town and doesn't have a lot to keep kids entertained.

We live in Ruidoso, which would overall be similar distance from Houston (took me about 10 hours getting home from CS this past Sunday). It's bigger than Cloudcroft and will have more lodging options and more to keep kids occupied outside of snow. It's a little lower elevation than Cloudcroft (7000-ish feet instead of 9000-ish) and thus a little less likely to have snow on the ground in town, but there is a great tubing area just north of town (Ruidoso Winter Park) that makes gobs of snow and can be really fun for the kids. Ski Apache outside Ruidoso is a much larger ski area than Cloudcroft has, but early January is very hit or miss on the quality of snow. It can also be a pretty gnarly drive to Ski Apache even when dry if you aren't used to windy mountain roads with no shoulder and limited guardrails.

If you come to Cloudcroft or Ruidoso, be sure to hit Carlsbad Caverns on the way here or way back. Roswell is a tourist trap and I wouldn't recommend going out of your way to see it. Lincoln is worth a day excursion if you like history, they've done a great job preserving historical buildings from the time of Billy the Kid. White Sands is a must do day excursion with kids, be sure to grab some round disc sleds (a lot of AirBnBs keep some on hand for their guests to use). Some other neat things to see in the area, but a little out of the way, would be Three Rivers Petroglyphs and Valley of Fires.