Don't doubt Walter Vinson told you that, but you have to look askance at it, don't you? There were a lot of ripoffs going on in the business in those days (and today), but everything that everybody claimed as theirs was only marginally so. The tune of this, for instance, was almost surely not WV's.
The Sheiks recorded SOTOTW in 1930 -- which is not to say definitively when it was written, of course -- but the tune had been recorded by Leroy Carr in "How Long Blues" in 1928 and by Ida Cox/Papa Charlie Jackson in 1925 for the very similar "How Long, Papa, How Long?"
Of course the tune was also used in Robert Johnson's "Come On In My Kitchen" and a number of other songs. I've got one version -- can't remember offhand who does it -- of the tune showing up with gospel words.