As long as there are no road closures on I-40 over the Blueridge/Smoky Mountains of NC and Tennessee, you should be good to follow it all the way to Flagstaff, then head south on I-17 to Phoenix. The highway remains open all year through New Mexico (where you will traverse the Rockies). Along this route you'll find lots of hotels, campgrounds, services, and places to fuel up. If wicked winter weather (blizzards or ice storms) make this route untenable, then take the southern route: I-85 from Charlotte to Atlanta, then I-20 through Texas to I-10. While I often recommend scenic routes away from Interstates, I personally wouldn't risk that while towing a camper across country.
Debbie