According to Wikipedia, "'Doxy' was written by Sonny Rollins during his stopover in England on a European tour. Its name is given after a bread-spread that the band was eating in the hotel." Here's the first recording of "Doxy," with Sonny and Miles (1954):
Although it wasn't the first recording of "I'm a Ding Dong Daddy From Dumas," one of the best is by Louis Armstrong (1930):
Here's another 1930 recording by Slatz Randall, posted as a 2-song medley (both sides of the record), with "Daddy" preceded by a tune called "Skirts," which has the same chordal template. You'll notice that Slatz's version of "Ding Dong Daddy" includes a lead-in section, sort of like the "verse" in many "Golden Age" standards:
But to get back to Louis - Here is a really great writeup by Armstrong expert Ricky Riccardi, with a wealth of interesting details about the song, Louis, Dumas (Texas), and much more.
In this article, the author mentions "How Come You Do Me Like You Do" (1924) as the source of the "Doxy" changes. Maybe. It's closer than "Ja-Da," but the bridge to "How Come" is still a bit different, ending on the tonic chord. I think "Ding Dong Daddy" is still a better fit. Or maybe "Skirts." Maybe all of them. You be the judge!