Sorry, "The earliest known copy" means that guitar arrangement. The earliest edition ("Gwine to Run All Night, or De Camptown Races") is:

Title: [Foster-Hall Reproductions]. Foster's Plantation Melodies. No.3. Gwine to Run All Night [Camptown Races].
Composer, Lyricist, Arranger: Written, Composed & Arranged by Stephen C. Foster.
Publication: Baltimore: F.D. Benteen, 1850.
Form of Composition: strophic with chorus
Instrumentation: piano and voice
First Line: De Camptown ladies sing dis song, Doo-dah! doo-dah!
First Line of Chorus: Gwine to run all night! Gwine to run all day!
Performer: As Sung by the Christy & Campbell Minstrels, and New Orleans Serenaders.