I've never made that road trip (at least not all of it), but at that time of year the route I would expect to take would be through Alabama to I20, then head due west. It will turn into I10 which will take you all the way to CA, I believe. Should be nice and warm. We both grew up in TX and I don't usually anticipate any bad winter weather there until mid-January. Doesn't mean it won't occasionally happen, but normally safe. If you head north before you get far enough west, I would think you could expect snow in the mountains but I wouldn't call if terrible weather as a rule. Good luck - safe travels!
2015 Legacy SR340 360RB
2014 Jeep Wrangler Unlimited Toad
Gigi, Poppy and Sadie Lady...On the road, but not full time!