I just did this trip for the sixth time last week. I prep for these trips by going to bed earlier starting 4-5 days ahead so I can go to bed on travel day in mid-afternoon for a late evening departure. I'm a bit further north than you (starting from central MA), but our total on the road time was 19.5 hours with about 3 hours off the road eating meals and stopping to stretch/use the bathroom and covered about 1250 miles total. We left at 9 pm. We stopped at the Clara Barton rest area just after exit 2 on the NJTP for gas and a leg stretch/bathroom break at 1:40 am which would be almost equivalent to your Philly starting point. We have an EZ-Pass, so the tolls from NJ to MD were a non-event. If you don't have one I'd recommend getting one.
We hit torrential rains in the DC area, but the eastern I-95 route is well drained and well lighted. We crossed the bridge into VA at about 4:45 am and kept going through VA on I-95 (Richmond traffic was light so we didn't take I-295). We crossed over into NC a bit before 7 am and used the nice NC Welcome rest area there for a leg stretch/bathroom break. I also took a power nap here that lasted about a half hour before a car door slamming nearby woke me and I was full awake. We stopped for breakfast at Cracker Barrel in Roanaoke Rapids, NC which was just a bit south of the rest area. We were back on the road by 7:50.
It takes just under three hours each to get through both NC and SC on I-95. Rest areas are well spaced for 5 minute stretch breaks. Speed enforcement was heavy on Wednesday 8/18 when we traveled through. We set the cruise to the speed limit for the whole trip and still made excellent time. We crossed into GA just after 2 pm and continued to just south of Savannah and had lunch about 20 miles into GA where an exit had tons of food choices and easy on/off.
We were back on the road about 3:15 pm and hit the only construction from exits 58 to 42 in GA. Speed was reduced to 60 mph for the stretch, but no backups slowed us. We drove straight through Jacksonville at evening rush with no slowdowns on I-95. We stayed right at SeaWorld and were at our hotel at 7:30 pm. Add 10-15 minutes and 8 more miles to get to Disney.
We made the trip in a 2009 Honda Civic Hybrid Sedan. It was fine for two people and a ton of college items and we only had to fill the tank three times enroute. My daughter only slept a couple of hours in the NC/SC stretch, but was otherwise good company as was the Sirius/XM channels for music and traffic as needed.
I was very tired at the end, but a 3 hour "nap" re-charged me and we went swimming about 10:30 pm and then had a normal night's sleep. I was not tired at all the next day.
Have a safe and fun trip!!