Vance Randolph called the song "The Dark Hollow," posted previously.
Compare with the version posted by Sandy Paton, above, also verses from the Hammond version posted by jon w.
Also related to "Free Little Bird," KY highlands, which shares some verses.

FREE LITTLE BIRD

I'm as free a little bird as I can be;
I'm as free a little bird as can be;
I'm as free a little bird as ever spoke a word;
I'm as free a little bird as can be.
2
Go bring me a chair and set me down;
Go bring me a pen and ink to write it down;
And at the end of every line that I write down
The tears will come falling to the ground.
3
If I were a little fish,
I would never bite the hook any more;
I would fly away down in the middle of the sea,
Where the bad boys cannot bother me.
4
If I were a little dove,
I wouyld fly from vine to vine,
And let you weep for your true-love,
Just as I weep for mine.
5
If I were a little bird,
I would never build my nest on the ground;
I would build my nest in sweet Kitty's breast
Where the bad boys cannot tear it down.
6
If I were a honey bee,
I would never steal the honey from the cup;
I would steal one kiss from my sweet Kitty's lips,
And I'd fly away to old Tennessee.
7
I cried all day like a child,
And I cried all day the day before;
And I ain't going to cry any more.

From Laura Lawson, coll. Harvey H. Fuson, in Fuson, "Ballads of the Kentucky Highlands," 1931, London, p. 130, without music.
In Duncan Emrich, 1974, "American Folk Poetry, An Anthology, pp. 110-111.