This reminnds me of a song, not a game, from my childhood, especially the line about going to the well to wash my toe / when I got there the water was low / what time is it, old witch, old witch, what time is it, old witch?
Old witch, old witch, she lives in a ditch
And combs her hair with a hickory switch
She lives on [various gross things]
And if you go near she'll [various rhyming horrors]
Then in the next verse you go to the well to wash something else and the water is something rhyming [hand / sand, etc].

The song starts, I think, Old woman, old woman, my [something alliterative that rhymes with the forthcoming washable body part).

The only weird Tag game I recall is Yellow Fever, where if It tags you you are It too but the children out of sight don't kbiw it. What a pandemic game, eh.