Another piece even earlier may have had some influence on both songs. It has no text in common with either but the ideas and sentiment are there. It is 'The Tyrannical Wife' in Merry Drollerie, 1661, which starts

It was a man and a jolly old man
       Come love me whereas I lay
And he would marry a fair young wife
       The clean contrary way.

He woo'd her for to wed, to wed
       Come...
And even she kickt him out of the bed
       The clean....

I have 4 slip copies of R-A-T from about 1800 all with significant variation, to me at least indicating that the song had been popular in oral tradition up to then. I'll email you them. Then I'll check my indexes for American versions but I doubt if I've got anything not in Roud or Bronson.