There are 2 distinct songs being put forward here. Admittedly they have become confused in some American versions and share tunes and refrains there. In Britain where they both originated they have very separate origins and evolution as initially broadside pieces, both quite early, 17th century at least if my memory serves me correctly.

The wife wrapt in wether skin is usually easy to identify by the wife-beating motif.

The Slattern Wife doesn't have this motif and is just a humorous catalogue of the wife's misdeeds, mostly concerning her neglect of the house, cows, and making of butter and cheese. The best known surviving version this side of the pond is probably 'Willy went to Westerdale'