Must be comming from pretty deep under ground to be that cold in TN. I've never had that issue on my RO unit here (it's also in my basement so I may just never have seen it, LOL), but my copper supply lines sweat like crazy if I run a load of laundry.
If you have space somewhere (possibly remotely if it doesn't fit under the sink) get a 5 gallon bucket fill it with water and put a coil of 100' or so of tubing in the bucket full of water, this is usually enough. If this still sweats you could use a small aquarium heater to maintain the bucket at a little above room temp. This actually will improve the function of your RO/DI unit as well as they work better with room temp water.
I've seen people use flow thru aquarium heaters activated by a flow switch also but that's alot more work and cost, but would save on space.