That isn't a version of Billy Boy, for all the similarity of the first verse. It's a form of The Rigwiddy Carlin (various spellings) which has also been found as Tam Booey (various spellings), Magherafelt Hiring Fair, and so on. It's number 366 in the Roud Folk Song Index. From what you quote, I'd think it's the set that Peter Kennedy recorded from Dicky Lashbrook at Kelly, Devon, in the early 1950s. It appears in Folksongs of Britain and Ireland (p 450) as Bargain With Me. More common in Scotland, it seems, where Peter Buchan published a text in 1828.