To download Billy Murray's version of "Alcoholic Blues," go here:

http://www.archive.org/details/BillyMurray_part4

It's about halfway down the page (the listing includes numerous Murray songs); there are two recordings, each in several formats (high bit rate, lower bit rate).

The Internet Archive has hundreds of Murray sides, comprising a fabulous collection of ragtime pop and novelty songs from 1900-1925. The Leadbelly take, while it's apparently labeled "Winnsboro," seems to get its lyrics from "Alcoholic" and is probably a version of the original pop hit, not the cotton mill parody.

And there's another recording by Vernon Dalhart, right here:

http://cylinders.library.ucsb.edu/search.php?queryType=%40attr+1%3D1016+&query=alcoholic