I know constant Billy as:

Oh my Billy, my constant Billy, when shall I see my Billy again?
Until the fishes climb o'er the mountains, then I shall see my Billy again.

Then the choruses go:

Billy again, Billy again, Billy gain, Billy again,
Oh, my Billy, my constant Billy, when shall I see my Billy again?

The only other Adderbury song I can think of goes:

Oh dear mother, what a fool I be!
Six young fellows came a-courting me.
Three were blind and the others couldn't see.
Oh dear mother, what a fool I be!

Unfortunately, I can't remember the name of the dance it goes with.