In 1929, Jimmie Rodgers recorded a quite different sentimental piece by this title - as far as I can see from a quick search, it too is different from those in the DT. It was written by Rev Andrew 'Blind Andy' Jenkins from Atlanta. Jenkins is said to have composed over 800 songs, most of them sacred. His two most famous were 'Billy the Kid' and 'The Death of Floyd Collins. In Jenkins' song, the drunkard's child had become an 'orphan child' by the third line even though it is not clear that dad had actually carked it. The song ends:

Don't weep for me or mother
Although I know it is sad
But try to get someone to cheer
And save my poor lonely dad

'I'm awful cold and hungry',
She closed her eyes and sighed
Then those who heard her story
Knew the orphan child had died

--Stewie.