There is small doubt that the song was written by Bill Cox. See the notes to his 1967 "rediscovery" LP, "Billy Cox -- the Dixie Songbird," in which he claims that the song was written at a New York hotel, where he and Hobbs were staying while in town to record. Nobody else covered the song, so it's not likely that a plugger was shopping it around to hillbilly artists. The next recording of it, I believe, was the one by NLCR and there may have been none after that till Cox's 1967 LP on the Charleston-based indie Kanawha label.

Cox was in his 70s when the LP was recorded, and he may have misremembered some of the chronology. He says that he and Hobbs missed the 1936 election because they were recording in NY, and that he wrote these songs to commemorate it, but Russell's discography shows that Cox and Hobbs recorded on 17 and 28 November 1936, so he was probably still in Charleston on Election Day.

He was a pretty facile songmaker, usually coupling clever lyrics to some older tune (or a variation thereof). The notes to his 1967 LP are forthright enough. He acknowledges traditional sources for a couple of the songs, and credits Riley Puckett and the Delmore Bros for others (he made several variations on "Browns Ferry Blues" during his recording career). The only demonstrably false assertion is that he wrote "Filipino Baby," actually a Charles K Harris pop song from 1899 (Cox was born in 1897). Cox made the tune more hillbilly, less ragtime pop, but the lyrics are little changed. (Cf Cox's remake of the 1905 pop song by Albert von Tilzer and George T Smith, "The Whole Dam Family.")

Someone who interviewed Cox in the late 60s told me that he had characterized his New Deal songs as "foolishness" and inferred a conversion to the Republican party, though the interviewer was a pretty rabid Republican and this may have been no more than a case of wishful hearing.