Because the Mudcat website was unobtainable most of today, Laurel Paulson-Pierce started a discussion of this song on the Mudcat facebook page.

There are three versions of Nickety, Nackety aka I Married Me a Wife in the Max Hunter collection.
https://maxhunter.missouristate.edu/songinformation.aspx?ID=657 (links to the other two songs are at the bottom of that page)

Laurel shared a recording by Chubby Parker https://www.youtube.com/watch?v=Rt2ZKLlbVTE

None of these songs include the verses about wife-beating which we find in The Wife Wrapped up in a Wether's Skin or The Wee Cooper of Fife.