Ha! Dialect foiled me again. (I've been doing this long enough; I should know better.) You've got to search for CLIMBIN'....

I listened to the performance, and yes, the sound quality is poor, but it seems to correspond closely to what you've posted, except that they didn't sing verse 4.

The sheet music was published by the Martin and Morris Music Company, whose archives are in the National Museum of American History. Unfortunately, the website gives no further information except that it was published between 1930 and 1985.

BMI says that I'M CLIMBING UP THE ROUGH SIDE was written by Thomas A. Dorsey, and the current publisher is Unichapell Music.