Cite that that’s really them? Sounds like a parody duo.
The Welcome to Duloc song from Shrek.
Please stay off - of the grass
Shine your shoes - wipe your… face
Weird Al’s great unrecorded song, “It’s Still Billy Joel To Me” uses this technique:
“It’s a big hit,
Isn’t it,
Even if it’s a piece of junk.
It’s still Billy Joel to me.”
And Bowser & Blue make it the entire point of their “Polka Dot Undies” song:
“The moral of this story, like a jewel it is gleamin’
But you will never find it in a glass of warm milk or tea…”
South Park
…Mrs. Roberts didn’t like him, but that’s because she’s a cunt-
-aminated water can really make you sick,
your bladder gets infected, and blood comes out your dick-
-tate what I’m saying…
The Magnetic Fields – Fido, Your Leash Is Too Long
“You scare me out of my wits when you do that shit-tzu” and “I don’t care what you fucks-shounds do”
A similar effect is the “ghost” rhyme as used in the chorus of I-Feel-Like-I’m-Fixin’-to-Die Rag by Country Joe and the Fish.
“And it’s one, two, three, what are we fighting for
Don’t ask me I don’t give a damn, next stop is Viet Nam
And it’s five, six, seven, open up the pearly gates
Ain’t no time to wonder why, whoopee we’re all gonna die”
(Just checking Wikipedia, apparently this method was from an old song. The author’s daughter sued Country Joe MacDonald for copyright infringement. Lost for filing way too late. Had to sell her rights to pay attorney fees.)
One thing’s sure, and nothing’s surer: the rich get richer, and the poor get children.
Yeah, just like it says right under the video: “Song parody - Bob Kevian and Tom Griswold (impersonating Dean Martin and Jerry Lewis)”
Benny Hill did this. In one monologue, he mentions being represented by the law firm of Martin, Barton, Parton, and Fargo.
A recent popular song starts with:
You make me come . . .
You make me complete . . .
You make me completely miserable.
Then there was a Don Knotts and Tim Connor movie, The Private Eyes, where the killer kept leaving mis-rhymed poems, including (and I remember this from more than 30 years ago?):
I am loose and free on the moor,
If you want to be safe,
Lock all your windows and screens.
Blondie, “Look Good in Blue”
“I could give you some head
And shoulders to lie on”
There’s also a lovely little ditty to the same basic tune, but much more risque: The Assumption Song. 
Yeah, I can’t see that on a Mitch Miller album!
Beer advert, (one of a series)
TVTropes calls this “Subverted Rhyme Every Occasion.”
From Curt Hennig and the West Texas Rednecks’ second hit, Good Ol’ Boys
We got an old hound dog and a pick up truck
We like long legged country girls that know how to… love
The totally sincere Tom T. Hall song I Love
“I love…Tomatoes on the vine
And onions.”
When Ellen DeGeneres was still being coy publicly about her orientation, she was singing in an episode of her TV show. She was looking at herself in a mirror, holding a dress up to heself, and singing “I feel pretty, oh so pretty, so pretty and witty and HEY!” as she saw something wrong with the garment.
Johnny Horton’s classic “The Battle of New Orleans” has the lines "Old Hickory said we could take 'em by surprise/ If we didn’t fire our muskets till we looked 'em in the eyes/ We held our fire til we seen their faces well/ then we opened up our squirrel guns and really game 'em… Well…
From the lads at the Reduced Shakespeare Company:
Othello loved Desi
Like Adonis loved Venus.
And Desi loved Othello
'Cause he had a big … sword.
I was a cheerleader for a goofy charity basketball game in high school. One of the cheers was:
we love wine, we love beer
but most of all, we love to cheer!
we love wine, we love cold duck
but most of all, we love to cheer!
Our parents didn’t appreciate that one. 
There was an Albanian monk,
Who slept in a very high bunk;
He dreamed that Venus
Was sucking his finger
And woke up all covered in sweat.
(Rolled 'em in the aisles back when I was a schoolboy)