I grew up in Eastern KY. We were in the historic Hatfield and McCoy feud area. It was in the late 70's early 80's we played this game with neighbor kids on the steps of an old church. There was an old outhouse on the property, made a perfect "well" for our witch in game!
I don't remember the "I'm going downtown" part, we played if I remember right that Mom was going to the store and we were not to get dirty.
When she returned she calls us for dinner, the kids would line up and she wanted to check our hands, she would come by and say dirty, dirty as she went down the line.
Then she would say to go to the well and wash your hands.......the person playing witch would appear and kids would scream and run back to the Mom.
She didn't believe their cries of "Witch in the well" and sends them back threatening to whip them.
Kids go again to "well" , witch appears, kids run screaming back.
Mom gets aggravated and leads them back to well, the witch jumps out chasing them all and whoever she catches is the witch next.
That is what I remember........we too thought it was a game passed to us from older siblings and that someone local had made it up.