January is the month voted most likely to have yucky weather in Tennessee.
February tends to be OK, though you can still get bad weather. We don't get a lot of snow, but tend to get ice storms.
March is usually OK, but we head somewhere warmer for our Spring break. A few times we've left a day or a few hours early because snow was coming in. If there was snow in March, it probably wouldn't last too long.
April is probably one of the best months to visit. The early flowering trees (redbuds then dogwoods) and bulb plants (tulips, daffodils) are glorious.
December could also be a good choice. We hardly ever get any winter weather that early and the Opryland Hotel would be nicely decorated. If you also visit the eastern end of the state, Dollywood has a few million lights, as does the town of Pigeon Forge.
My biggest concern for you in Jan - Mar would be the weather you might encounter between here and NY.
Sheila