Dick G.: I think, in fact, "tease rhyme" is the technical term.

This girl had a sister -- I blush to confess,
She once met a man and began to un-
Nerve his by zipping her dress down the front,
But only to give him a look at her
Cunning gold necklace that she always wore,
But he grew afraid that she might be a
Horrible schemer and up to some trick,
But she said "I see that you have a hard
Problem you might take in hand if you tried",
So he stepped up to her and put it in
These words: He said that when he lived in France,
The ladies he like let him into their
Plans, and would tell him what they'd like to do,
So she said "Stop talking, 'cause I want to
Scratch my mosquito bites right where they itch",
And he said "Well, I am a son of a
Bishop, but I'll be a layman instead",
So he picked her up gently and laid her in
Sweet violets, etc.