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Does the car without snow tires at least have a good set of all-season tires on it? If not, buy a new set. You won't need snow tires when you get to TN, but you do need all-seasons.

The "best" route would be the most direct route using Interstate highways and avoiding known trouble spots. In this case, this would be I-90 to Sioux Falls, I-29 to KC, I-435 bypass around the NE side of KC, I-70 to Wentzville MO, I-64 to I-57 to I-24. As long as Snoqualmie Pass is open and reasonably clear this route avoids most high elevation road issues. I would not take I-82 to I-84 to I-80 because I-80 across WY is notorious for winds, whiteouts, and black ice. This is a 5 day drive in decent conditions. You need to keep an eye on weather and road conditions (a good site for road conditions everywhere is safetravelusa.com) and be prepared to find a hotel to wait things out and/or find a reasonable detour.