anabelle,
Looks like I'm a few days late on this but here's my two cents (I live in Knoxville, by the way):
1) Yes, it does get that chilly here in January/February. I don't think I would want to tent camp at that time of year either. As far as hotels in the Knoxville area - you'll be able to find all kinds of nationally known hotel chains right at interstate exits. (Specifically you'll find these hotels where I-75/275 & I-640 intersect, and where I-40 & I-75 run simultaneously) Knoxville is the crossroads of 2 major interstates that span nearly the entire country going east/west (I-40) & north/south (I-75, which is what I assume you'll be taking). Since Knoxville isn't necessarily a "tourist town," and the hotels are mostly hoping to attract those passing through, such as yourself, I think you'll find the rates pretty reasonable.
2) Pigeon Forge/Gatlinburg is a great place to visit but I wouldn't recommend it in YOUR case. I agree with previous posters that you will have to drive about 30 minutes off course, get off the interstate, and then potentially drive another 30-45 minutes to get to your hotel. All in all, we'd love to have you visit Pigeon Forge/Gatlinburg in the future, but I don't think it fits your needs right now.
3) If, in the future, you are going to be passing through at a warmer time of year and will be looking for a campground in Knoxville I will also recommend the Escapees RV Park on Raccoon Valley Road. DH's aunt & uncle recently stayed there and it was clean, the people were friendly (of course) & it was less than 5 miles from I-75. His aunt told us that all Escapees Parks are loosely associated with each other -- maybe they're privately owned but they all hold to certain standards?
I hope that's helpful. Please PM me if I can give you any Knoxville-related info!