We've driven a few times to Fla, and my parents drive it every year as they have a winter place down there. Their preferred route is the I-90 to -77 to -79 to -26 (IIRC??) then -95. They've driven both, and find the more eastern route more senic, and because you don't have to drive through any really big cities (Charlotte would be the most major city) you avoid traffic snarl-ups, rush hour etc.
Also, as someone mentioned you can take the US-19 bypass that links IIRC I-77 and -79 together, but we missed the cutoff coming home one year (following GPS and not paying attention to road signs) and it didn't really add any appreciable time to our trip, and was still quite scenic.
I personally don't like the I-75 route because you have to drive 3 hrs west to Windsor before you make the cross over to then head SSE for the rest of the trip. Driving the east route you're driving as straight a route to Disney as you possibly can.
Also, unless you aren't great for long stretches in the car, doing the trip over 3 days seems a bit much. From Hamilton it SAYS it is a 20 hour drive time.... but we usually do the actual drive time in about 17-18 hours total. My wife does all the driving, and we don't speed any more than maybe 5 miles above the speed limit.
When we have done the overnight stop we usually do so around Columbia or a bit further south along the route so that the second day in the car is a short one, and has worked out really well for us. Last time we drove we actually found that we were hitting our regular stop place about 7AM (we start the drive at night so our kids sleep through a good 8 hours of the drive and makes things so much easier) and so decided to push on to see how much further we could go, since we were almost in Georgia at that point. Suffice to say we did the trip in a straight drive and it was pretty easy...so we're going to try it again this year.
I don't recommend the drive straight thru, it isn't for everyone, but a one overnight stop is easy to manage (it is what my older parents do all the time). Should also mention that we have never had problems with traffic, speed traps, finding a place to eat/stay/gas along the route either.
Good luck in whichever route you choose BTW!!!
Tim