In Kentucky, in my family, this game-song was called, "Old King Cole was a Jolly Old Soul- goes like this:

Old King Cole was a jolly old soul
And that you may know by his larnin'-
He eat corn bread till his head turned red,
And his old yeller cap needs darnin'.

My pretty little pink, I once did think
That I and you would marry.
But now I've lost all hopes of you
And I ain't got long to tarry.

I'll take my knapsack on my back,
My musket on my shoulder-
I'll march away to Mexico,
Enlist and be a soldier.

Where coffee grows on the whiteoak trees
And the rivers they run brandy;
Where the boys are pure as a lump of gold
And the girls are sweet as candy.

You may-go on and I'll turn back
To the place where we first parted;
We'll open up the ring and choose a couple in-
And we hope they'll come free-hearted.
    © 1953, Jean Ritchie Geordie Music Publishing Co.
       (from Balis Ritchie, Jean's fath

My dad had another "little miss" song, like this:

Where are you goin my pretty little miss,
Where are you goin my daisy:?
Well...if I don't get me a young man soon,
I think I'm goin crazy.

Cho: Hi rinktum a dinktum a diddle diddle dum
    Hi rinktum a dinktum a doodie (repeat these 2 lines)

How old are you my pretty little miss-
How old are you my honey?
Well...if I don't die of a broken heart
I'll be sixteen next Sunday.

Then there are several more verses. Would this be the one you're looking for, Frogmore?