So after we finished up in the library, we headed over to the city cemetery to see if we could find his grave. The cemetery was fairly large. According to the catalog we were reading, he was listed as being buried in the "New Addition" section. It took about twenty or thirty minutes before I happened upon the main head stone with the name Dunaway engraved on both sides.
Near the headstone were three flat markers with the following names: Sid Dunaway, Ollie H. Dunaway, and Junell Birmingham Dunaway.
Without getting into too much detail, Sidney Dunaway appears to have been Dewey's older brother - at least according to my research so far. Ollie was Sid's wife, and Junell was apparently their daughter, who died in her childhood. Sid's name is also seen on other relevant documents, the informant on Dewey's death certificate being one example.
Other details -- the 1920 census shows Dewey to have been living in the Palestine area with his daughter Margaret and wife Julia Mae. Also, his death certificate says that is occupation was as an "insurance man", and a city directory from 1935 gives his occupation as a salesman. So there does seem to be some consistency. This directory also shows Sidney, Ollie and his mother Mary, widow to J. A. Dunaway (another mystery for another time), living with Dewey at this same address.
Dewey Dunaway died on December 19, 1936 in Palestine, Anderson County.
I'll definitely be re-visiting my great grandfather as soon as new reliable evidence makes its way into my hands.