There is a "Fly Around My Blue Eyed Gal" which has the same tune. And a "BETTY ANNE" to the tune of "Shady Grove," with a couple of verses apparently lifted from Fly Around etc. Threads on the song are HERE and HERE and HERE and HERE

The song scans the same as "Shady Grove," so that is an easy matter to lift verses to the one and move them to the other. I think it's going a bit far to conclude that because verses from one song crop up in another the songs are the same. Especially where it's possible the person you just heard sing the song could have been the very first one to throw in those particular verses. Let's see:

Who's going to shoe your pretty little foot,
Who's going to glove your hand
Who's going to kiss your pretty little foot,
Who's going to prove that this pretty little song,
and all the 1327 others which have this verse in it, are all the same as Shady Grove? Not me!