In the winter the scenic route or the shortest route is not really the best route. I have found that you can change your departure date if bad weather is forecast, but once on the road you can run into a problem based on unforseen or unpredicted weather.
My suggestion is long but safer. Go due south as quickly as possible at the lowestest elevation. I would consider I-15 to Las Vegas, then US 93 to Phoenix and then route I-10 east thru El Paso, then US 90 East (which is more interesting than I-10).
