A lot of songs have the meter of "Old Joe Clark-Liza Jane." One of these is known as the "The Gal From The South," which usually includes the line

"Old massa had a yaller gal,
He brung her from the south," etc.,

It includes this verse:

Old massa had a banjo,
Its strings were made of twine;
The only tune that I could play
Was "I wish that gal was mine."

"My Liza Jane" has these lines:

Saturday night and Sunday too, pretty gals on my mind;
Monday mornin' just 'fore day, white folks got me gwine,
White folks got me gwine, white folks got me gwine.

When I go to my gal's house and she is at home,
I lays myself back in de big arm cheer and picks on de old banjo,
And picks on de old banjo, and picks on de old banjo.

As I went down the new cut road, she went down de lane.
Was de last time I saw my true love, so go 'long, Liza Jane.
Go long, Liza Jane, go long Liza Jane.

Not what you want, but offered for comparison. From Newman L. White, American Negro Folk Songs. The chorus you offer is common to several versions of Old Joe Clark (in Lomax, etc.) but not with verses suggesting slavery or playing the banjo.

In Randolph, Ozark Folksongs, is the verse:

I had a banjo made of gourd,
The strings was made of twine,
But the only tune that I could play
Was I wish that gal was mine.

No verses in Randolph suggest slavery; this one, also from Randolph, suggests prison:

Old Joe Clark is dead and gone, I hope he's a'doin' well,
He made me wear the ball and chain,
An' it made my ankles swell.

No other verses with this fragment collected in 1930.