Katarine - We moved to WNC from East Texas and in our many trips back and forth, we found the best route from Alabama to be I-20 through Atlanta, then catch I-85 North to I-26 West; you'll be getting on I-26 just north of Spartanburg, just about 18 miles to the NC line. It will bring you right to the Hendersonville/Flat Rock, NC, exit (#53) that is close to Dupont. That way you're on a major multi-lane highway all the way, easy access to gas (roomy truck stops) and for emergency help (God forbid you need it). The only big city you'll hit is Atlanta and it's not really too bad if you time it so you're not hitting a main rush hour. I-85 can be a pain with traffic, depending on day and time; just be patient and you'll be OK. You might be used to traffic like that in Birmingham, I don't know.
As for the Brevard route, there are some highways there that are 2 lanes, curvier, more rural so less gas and fewer places to pull off if you have to. If I were on a scenic road trip I'd try that way, but not with a big horse trailer. Just my opinion.
FYI, Dupont has several riding trailheads so specific directions within the Forest will depend on your actual destination. If you have time before you leave, order the great trail map that Dupont has; you can get a laminated type that's pretty indestructible. You'll see the trailheads by name plus all the riding trails.
It's just grass and water till it hits the ground.