Well if you're at 3,500ft,quite a bit lower than Denver.Denco is 5,280ft,mile high. The northern 240 miles or so of NM are above 7,000ft to the CO border then you dip down and back up into CO Springs @ 5,500ft or so then down a bit into Denver.
Albuquerque is at 5k and the surrounding Sandias crest @ about 10k to the east. Snowfall varies greatly by year but the Jan temps are easily in the teens or below every night.
Down south where I am we have mountains to the far east and west. We have the Gila National Forest to the west and Ruidoso,close to the Mescalero Apache Indian Reservation to the east. I don't know much about the Gila. I've never been there! Lived here most of my life and never gone. I'm upset with myself about that. Now near Ruidoso is Sierra Blanca,I think 9k and you can ski there maybe three months out of the year if we're lucky. From what I know the state mountain ranges don't have the drastic temp changes as you describe or like what one may find in Colorado. We definately get snow in our mountains. The desert can get weird cold as well. I work outside from October to January and it gets down to the teens. It's just a different cold though. I could handle 5 and below in Denver but it just seems colder here. Sadly I think I could tell you more about Colorado mountain climate than my own state! They had a saying there "if you don't like the weather,wait ten minutes". It's true. In the foothills around Boulder and Denver you'd be amazed. One year they had a fire threatening homes near Boulder in August,temps were in the low 90's and really dry. The night of the day they were advising evacuation it snowed and put the fire out. Crazy