Buy some standard snow tires. Avoid hydrophilic snow tires (Blizzaks) - these wear excessively on warm dry pavement. Get an actual snow tire, not a damn all season tire - all season tires suck big-time in any significant snow. Get 'em on all 4 corners.
Standard snow tires work just fine in any weather. Even in the 1960's, 70's, and 80's, my Dad used snow tires all year around. We went on vacation with snow tires on the car, traveled thru the Mojave Desert, Great Salt Lake area, etc and nowhere-near legal speeds, and had no problems whatsoever with tires. The snow tires will take anything you want to dish out to them. I have All Terrain tires on my Jeep Cherokee, and use them summer and winter, and regular snow tires before those, and this is (hot) Virginia.
The only thing you should maybe reconsider is using I-81 to get to Florida. I-81 is full of REALLY numerous and nasty truckers that tailgate the hell out of you, zoom down the mountain grades at 90+ mph, and cause numerous deadly accidents on that road, at least in Virginia. If you can get to I-95, and arrange to get thru the DC area at maybe 4 AM to avoid the high traffic density, you should have a much safer time of it. You'll get a less likelihood of big snows, too, as opposed to I-81 which has big snows and maybe even worse, fog.
Dave Head