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Since u are on the west side the mountains, I would (and do, and have made the trip a number of times, as we have family in Newport) take I-75 to Florida. There are a number of easy routes to the east coast, once you arrive, and the same with going to visit the rat. Just time your trip to make the best time thru Atlanta. With our Class A, we plan for 12 hours of driving and stop after about 6 hours (300 miles).
FYI, when planning your trips, always plan at 50 miles per hour. Free free to pm me for any questions u may have.
To avoid the mountains:
From Pigeon Forge, drive to Knoxville and pick up I-75 South (It will be congruent with I-40 W for awhile).
Try to time your trip through Atlanta for a Sunday before 1 pm for the lightest traffic. (It's almost deserted on their beltway at that time). Stay Saturday night just outside of the beltway and you'll be able to hit that sweet spot for little/no traffic.
First law of science: don't spit into the wind.
Keep on rollin'!
A few years back I was on the road over the mountains with the TV only. I saw a few dp's, gassers and a couple of good size fivers. As a flatlander from FL,
I personally would not want to go over the mountain with any type of RV. Have gone both ways around and I-75 as suggested above is the easiest to me. Have also done I-40 to Ashville, and I think it is I-26 down to I-85. I-26 had some steep downgrades and wasn't too thrilled with that.
I have a little story... when we were brand new to rving we came over part of that mountain from the NC side.... about the scariest drive we have ever done.... roads very steep and winding, it was a foggy morning and we couldn't even see the road a few ft in front.
We did survive and the trucks behind us were very patient... but i would never go thru those mountains again with the trailer.