To hear The song played in Minstrel style with the original words, there are bands that specialize in the minstrel music that mostly play and market their music through the Civil War re-enactment circuit. Two recordings that contain this song as it was first played are:

Joe Ayers-"Old Dan Tucker"
The Amoskeag Players-"Minstrels Of Great Reknown"


Labels unknown. Ayers is painstakingly authentic. From tapes someone gave me back when I reenacted. I guess you'd Google or go to the tents in the sutlery at your next local reenactment. I've been out of that scene for some years now, so there's probably more stuff like it issued since that's floating around.