“Rollin’ in My Sweet Baby’s Arms” by Flatt & Scruggs (and a lot of other people) kind of fits the pattern:
Well I ain’t gonna work on the railroad,
And I ain’t gonna work on the farm.
Gonna lay around the shack
Till the mail train comes back
Rollin’ in my sweet baby’s arms.
“The first line traditionally introduces a person and a place, with the place appearing at the end of the first line and establishing the rhyme scheme for the second and fifth lines.”
The form’s popularizer, Edward Lear, always followed this. As do all “joke” limericks (e.g., “There once was a man from Nantucket…”).
Apparently, the word limerick is “commonly” used more loosely now, so it turns out you may be right about that. I should have, it seems, used a different word, like “traditionally”.