Arizona/New Mexico in the winter--lots of questions!

You might want to check out Weatherbase. A lot of the cities and towns along I-40 get significant snow in January and February. Maybe drop down to I-20.
I had a friend that went to college in Flagstaff. They got a fair amount of snow each winter. They used to head for Tucson on weekends.
On that basis, you may run into RV parks that are closed or have water turned off. Or not. It isn't like you're driving thru North Dakota and Montana.