Well, here's the tune from the Frank C. Brown Collection. It doesn't sound like "Little Brown Jug" to me, but it does sound like a tune played at the end of a cartoon. I posted my transcription of Dancealot's tune in the message above. The two tunes are more-or-less the same. Very familiar, but I can't put a name to it.
-Joe-

Lyrics to this tune:

Oh, dear doctor, can you tell
What will make poor Mary well?
She is sick and she might die
That would make poor - - cry.

Music notation is given at The Frank C. Brown Collection of NC Folklore: The Music of the Folk Songs By Frank C. Brown, Duke University Press, 1970, page 540.

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Name that tune!!!