I hiked a normal, leisurely pace of doing the Trail in 6 months, from July 3 - January 11, taking a week off over Christmas. Winter in the Southeast is much more about cold rain than about heavy snow. There are occasional heavy dumps of snow for which one might want snowshoes but freezing rain is much more common. MicroSpikes work well for that. It's usually possible to hop around and avoid the ice entirely, but that gets tedious after a while.
When I went over Max Patch it was 70 degrees. When I was going over Roan Mountain, it got down to zero. The snow was only about an inch deep, though. Starting through the Smokies not long before Christmas it was warm and sunny. Finishing it was raining so hard that the M&Ms in my deepest inner pocket got soaked and the colors ran. Don't think you'll only have true winter weather--you won't. But occasionally a few days of it will happen.
You'll be fine.