There are a number of songs, with similar choruses; may or not be related.

The theme of 'another man's baby' is popular in writing, but I can't find songs related to your fragment except a couple of current ones of little use.

Not related in theme, but interesting:

Lyr. Add: CRADLE SONG

O Lulie, O Lulie, if you please,
Let me fall upon my knees;
Rock de cradle,
Rock de cradle,
Rock de cradle, Joe.

Joe cut off his big toe
And hung it up to dry.
All de gals began to laugh
An' Joe began to cry,
Rock de cradle,
Rock de cradle,
Rock de cradle, Joe.

Texas and Arkansas. Lyrics only, pp. 153-154, Dorothy Scarborough, 1925, "On the Trail of Negro Folk-Songs."