You didn't say just where in NC you are starting from but, in general the difference between traveling I10 and I20 will be around 125 miles.
I doubt the weather will be an issue either way although it's possible. Dallas and points east had a scare just prior to Thanksgiving. Houston and I10 didn't but I've seen some ice storms west of San Antonio on I10 that shut everything down.
I'd say the I10 route is a bit more interesting (Gulf coast from the Florida panhandle through the Houston area), then San Antonio and the Hill Country on I10. Then a lot of open country to the junction of I20/I10. On I20 you'd have Shreveport, Dallas, Abilene, then many miles of West Texas to the I10/I20 junction at Van Horn.
So it's your call. I10 is longer, probably will have better weather, and passes through more interesting places. I20 is shorter and the weather probably won't be a factor.
I'll be interested in seeing you post what you decide to do.