We have done both, and there are pros/cons to both. We drove from New Brunswick, so about 28 hrs vs your 19 hours. Kids were 5 and 8. Driving saved us a LOT of money.
We took 3 days, got phenomenal hotel deals using hotelcoupons.com (didn't pay more than $65 US and the hotels were clean, comfy, had full hot breakfasts included and pools), packed lunches and just ate out for supper.
We all LOVED the drive down. We did the route 81/77 through the Virginia mountains. It was beautiful, and we stopped to tour the Shenandoah Caverns. Honestly it was just so neat to see everything as we drove down.
However, the drive home was a LOT harder. We were all tired, the anticipation was gone, and it was more of a "I just want to be home" feeling.
In hindsight, DH and I have said that if we were to do it again, we'd push it on the way there and get there in 2 days. Then on the way home, plan 3-4 days, and probably do some sight seeing, something to a) break it up so you're not driving so hard when tired and b) something to just keep the interest alive.
Since then I have flown down with the kids, and I have to say, we LOVED it so much more. Sooooo much easier, though I personally just love flying (take off is so fun
) That said - we had an easy flight experience without any hiccups or delays. So that colors our judgement. If I had to deal with the hassles of delays, cancelations etc... I might be telling you I enjoyed driving more. You definitely have more control driving. What I didn't like about flying was not having our own vehicle - I far prefer having my own transportation, and it was too much money to fly, and rent a car.