Note: The name "Liza Jane" was a common female name used in many traditional songs from the US. Unfortunately, many of these songs were also rooted in minstrelsy. Blog about minstrel tunes used in music education by Robin Giebelhausen. Book about songs with "Lil Liza Jane."
Version from Jill Trinka's The Little Black Bull.
In their blog "Pancoco Jams," Azizi Powell documents several examples and variations of the song. Here is another source, from a dulcimer website called "Hogfiddle" which strongly associates this song with the state of Illinois. The Hogfiddle blog cites "Folk Songs and Singing Games of the Illinois Ozarks" as a source, however it is not available digitally and the cost is rather high.
I found another David S. McIntosh text, Singing Games and Dances, that has the song.
The Traditional Tune Archive documents the fiddle turn version of the song - there is a link to a recording on the site so you can hear how it compares to the original.