I also had the notion that "gay and grinding" might be a corruption of "grist and grinding" or better still of "grain and grinding". It seems that most versions of Child 10 include the part where the body turns up in the mill pond, so a refrain emphasizing the mill seems like a natural. I note that of all the versions of this song included in the links at the top of the thread, "The Two Sisters (8)" is the most similar ((not counting "The Two Sisters (12)", which is the same)). It was collected from oral tradition in Newfoundland. The first verse runs thusly:

There were two sisters, Jane and Mary Ann
I went a-gay and a-gandy
They were both in love with the same young man
Down by the bonny busk of London.