I decided to ask Old Crow Medicine Show what they meant. And they wrote back!

Rounder,....American English. in it's origin, the
word describes an itinerant preacher, who
"makes-the-rounds" the term usually being applied to a
methodist.
Then it came to mean any sort of transient worker.
Then it began to mean a drunkard. All in the space of
150 years it went from preacher to drunk. Tell David!
I bet he'll get a kick out of it.
Ketch Secor
OCMS