Third option is to get off I 10 at El Paso and take 180 east through Carlsbad to Weatherford. If you do want to go that way, take the Americas Blvd/Joe Battle exit on the east side of town and take the loop north to the Montana exit. Guadalupe Pass 80 miles east of El Paso is steep for about 7 miles and has high winds on windy days.
In the time frame you are looking at you may have icy roads to contend with going through Cloudcroft (8800 ft) or Ruidoso (about the same over Apache Summit).
There is a construction project on 10 between Las Cruces and El Paso where a bridge is being replaced. Traffic is choked down to one lane each way for about 3 miles.