Expressively drawn advertisement for this record from the Chicago Defender for July 19, 1924 uploaded, confirming the order of sides and release month as presumed.
I found an article in the same issue of Talking Machine World, written in Baltimore, MD, on July 10, 1924, that proves that this record, despite the advance information "for August, 1924", was available in early July, and already sold out in Baltimore. I have linked the whole article in the notes and uploaded the column referring to Bessie Smith here.
As requested, I've added the link to Talking Machine World. As already implied in my comment on OKeh 4253, I have chosen an earlier release date than the "advance" catalog month.
I have also flipped sides, which had been added in the same order as on 78discography.com. TMW, however, lists them in the order of the nearby matrix numbers, which may be a rule if we have nothing else, see xiphophilos' comment on Columbia 1-D.