Flattest, warmest route to Las Vegas in January from NC

I-40 will definitely have more hills. While being the shortest it will also be more likely to have winter weather issues in January.

Consider dropping down to I-20 and follow it to I-10 in west Texas. That route will be flatter than I-40, and shorter than an all I-10 route. Then keeping an eye on weather forecast. If winter storms are forecast for any where along I-20 then there are many places to drop on down to I-10.