Meade notes that there is a related text, 'Dat's Me and Marthy Jane', found in 19th century songsters with the chorus:

Then far (sic) you well Melinda, far (sic) you well, my Jane
Fare you well my Susie dear, I won't see you again

Grandpa Jones recorded 'Melinda' in 1950 (King 912). Fiddlin' John's was the only old-time recording of 'Do Round My Lindy' - recorded ca December 1925 in NYC and issued in March 1926 as OK 45032.
[Info from Meade, Spottswood and Meade 'Country Music Sources']

--Stewie.