Darling Little Joe: there are two sheet music printings, one (dated 1876) crediting it to Charles E. Addison, the other (1866) by V. E. Marsten. See Marsten's at Levy site:
http://levysheetmusic.mse.jhu.edu/levy-cgi/display.cgi?id=105.044.000;pages=4;range=0-3

Randolph collected several versions and the song entered tradition so the likely source of the Carter's 1939 version is Bradley Kincaid, who did the song in 1934.

BROADSIDES: Levy 105.044, "The Death of Little Joe," G. Andre & Co., Philadelphia, 1866
LOCSheet, sm1876 10660, "Little Joe," Blackmar & Finney (New Orleans), 1876 (tune)
"The Death of little Joey" H. De Marsan, Publisher, 60 Chatham Street, N. Y.

http://memory.loc.gov/cgi-bin/query/D?amss:2:./temp/~ammem_wo0t::@@@mdb=mcc,gottscho,detr,nfor,wpa,aap,cwar,bbpix,cowellbib,calb

Broadside LOCSheet sm1876 10660: "Composed and sung by Maj. Chas. E. Addison the noted Confederate Spy and Scout of Gen. John H. Morgan's Command."

DARLING LITTLE JOE-Carter Family

What will the birds do mother in the spring
The little brown birds around the door
Will they fly from the trees and tap at my window
Wondering why Joe wanders out no more

What will the kitten do mother all alone
Will it stop from its frolic for a day
Will it lie on its rug by the side of my bed
As it did before I went away

Keep Tyke dearest mother my poor little dog
For I know that he'll miss me too
Keep him when old and useless he grows
Sleeping all the long summer through

Show him my coat mother so he'll not forget
Little master who will then be dead
Speak often and kindly of little Joe
And pat him on his curly head

What will Thomas the old gardener say
When you ask him for a flower for me
Will he give you a rose he has tended with care
The first fairest bloom of the tree

And you dearest mother will miss me for a while
Though in heaven I'll no larger grow
Any kind angel will tell you at the gate
When you ask for your darling little Joe