If you still need to find 'another song' this one would at least predate the American Cousin and thus be plausible.

Dim recollection that one of the Williamsburg songbooks (source for the ballads sung in the colonial taverns at Colonial Williamsburg) had a charming and easily learned song, here's the first verse lyric:

Two maidens went milking one day,
two maidens went milking one day,
and the wind it did blow high
and the wind it did blow low,
and it tossed their pails to and fro, fa-la-la,
and it tossed their pails to and fro.