The song has been widely collected in America. Sharp has 18 versions of it, but no notes. It was collected in Florida too.

Cox (Folk Songs of the South as "Young Ladies") says that "it is also called "Little Sparrow" and states that "it is somehow related to "O Waly, Waly gin Love be bonny"", (Child 204 - Jamie Douglas) "printed in the early part of the eighteenth century, but even then regarded as old"

How related looks moot to me. Cox relates verse 3 of the Scottish song to a verse in Fair and Tender Ladies, the (Scottish) verse being:

    O waly, waly! but love be bony
    A little time, while it is new;
    But when 't is auld, it waxeth cauld,
    And fades away like morning dew.
  

I haven't time to compare them more fully just now, but I'll have a look later if I've got time.

Mick