I didn't know this song had clean lyrics. Note that it has nothing to do with limericks, and the tune is nothing like the one that begins, "Ay-ay-ay-ay." Evidently only the title got borrowed and stuck in another song.

Lyrics below from The Lester S. Levy Collection of Sheet Music:

WALTZ ME AROUND AGAIN WILLIE
(Words, Will D. Cobb. Music, Ren Shields. 1906.)
"Successfully sung by Blance Ring in Charles Dillingham's production Miss Dolly Dollars."

1. Willie Fitzgibbons who used to sell ribbons,
And stood up all day on his feet,
Grew very spooney on Madeline Mooney,
Who'd rather be dancing than eat.
Each evening she'd tag him, to some dance hall drag him,
And when the band started to play,
She'd up like a silly and grab tired Willie,
Steer him on the floor, and she'd say:

CHORUS: Waltz me around again, Willie; around, around, around.
The music is dreamy. It's peaches-and-creamy.
Oh! don't let my feet touch the ground.
I feel like a ship on an ocean of joy.
I just want to holler out loud, "Ship ahoy!"
Oh! waltz me around again, Willie; around, around, around.

2. Willie De Vere was a dry-goods cashier.
At his desk he would sit all the day,
Till his doctor advised him to start exercising,
Or else he would soon fade away.
One night this poor looney met Madeline Mooney.
Fitzgibbons then shouted with joy:
"She's a good health regainer. You've got a great trainer.
Just wait till she hollers, my boy." CHORUS.

[The Virtual Gramophone has a recording by Billy Murray and the Haydn Quartet, c1908.]