I learned the song with the name Annabelle and not the name Alice. This is the way that I learned it:

Annabelle, where are you going?
Upstairs to take a bath,
Annabelle, your legs are like two toothpicks
Annabelle, you neck is like a girrafe
Annabelle, pull out the stopper
Annabelle, please drain the bath
OH MY GOODNESS, OH MY SOAL, THERE GOES ANNABELL DOWN THAT HOLE!
ANNABELLE, ANNABELLE, glug.