Sadie Green, the Vamp of New Orleans, as recorded by the Jim Kweskin Jug Band back in the mid-60s. (I looked in the DT to see if this song was in there, but no luck.)

Or, indeed, anything recorded by the Kweskin, or any other, Jug Band, in the 60s, or in any other decade.

When I first saw this thread's title the other day, the very firsat song that came to mind, of course, was Jesse Fuller's San Francisco Bay Blues ~ which, of course, was also the first kazoo-friendly number that the OP had thought of, as well...