If you've read Byron's fishing report, then you know that floating anywhere in the mountains is even less likely in December. You could go over to the Tuck maybe but that's pretty much it for the winter. that's not to say that winter fishing is a bad thing... just don't think about floating in the park.
So you could wade fish the park and float any of a number of regional tailwaters. The Clinch, the Watauga, the Caney Fork...