Since this is a reference question, I will post more from work.

The Walter Dean Myers version (for it was he) quotes no source.

The Wolkstein version has only the note:

"Nat King Cole first heard the story of the buzzard and themonkey from his father, a Baptist minister in Montgomery, Alabama. He then recorded 'Straighten up and fly right,' based on the story, in 1943. [Her is a part of that song.]"

I have the impression us honkies only know the story from the NKC song. I did an extensive search some 15 years ago and only came up with that one paragraph I mentioned earlier. I will post it when I find it in my copious but disorganized files.