We have been in Louisiana on or near Interstate 10 many times in January. I think that December and January are the worst months for winter in Louisiana and southern Mississippi, which 10 will also take you through.
It is colder than you might think, and you could run into light snow or ice. We sat through an ice storm last February near South Padre Island, though, so no place is truly safe.
We didn't have any problems near the Mexican border, just behave yourself as the Border Patrol are everywhere.
If you have not spent any time in Louisiana and southern Mississippi, those are beautiful, beautiful parts of the country and would be worth your time to peruse thoroughly.
We like Gulf Island National Seashore in southern Mississippi.