We can never be sure what the original story was. There are so many known songs where what appears to be the main point has been lost in later versions. Perhaps we need a word to describe this "decay or loss of meaning" (or is there one?

Look at the Crabfish (or Crayfish, or Lobster)

The essence of the story is that the fisherman brought home a fine cran, and wanted to put it somewhere safe for the night. He decided on the chamberpot. In the night, his wife used the pot, and the crab grabbed her in a 'sensitive place'. When he went to investigate, it grabbed his nose with the other claw, leaving the two connected in a rather unusual manner.

"Alas quoth the good man, that euer I came hither
he has ioyned my wiffes tayle & my nose together"

But in later versions (such as Mr Razzledum), the second, more indelicate, part of the joke has been lost, and the fisherman just beats the crab to death.

Another example of this 'decay of meaning' has been seen in a joke called "The origin of the yodel"

This long story concerns a traveller who begs a place to stay on a cold night, and is told that he can sleep in the barn, provided he doesn't touch the farmer's daughter. He evidently doesn't keep his word on this, and in the morning, the daughter complains to her father, who rushes after the traveller, shouting
"Come back, villain, you've f***ed my daughter"

To which he replies "And your old lay–hee-dee"
(say it out loud to get the idea)

Now the point of repeating this is that I have heard the joke told with the punchline changed to "And your old woman too", which some how doesn't quite sound like a yodel.