Thanks to those who took the time to carefully read and respond to the specific question I posed.
Here is the additional bit I alluded to in my first message:
Virtually none of my family on this side lived in Atlanta in the mid-1930s. Most either lived in far southern Georgia or northern Florida, with a handful of branches outside of those two states. The only occasions when I think it was typical for them to be in Atlanta were when they were either there briefly on business (they may have made buying trips there to stock their clothing stores, although my sense is that it was more typical for them to go to New York for this purpose) or when they were there to attend college.
There is one member of my family who I know lived around there at this time. She was born in 1909 and sometime in the early 1930s, she and her husband moved to a town about 35 miles outside of Atlanta. From my prior research, I knew that in 1936, she had childbirth-related complications (eclampsia) from her first delivery that eventually caused her death after a hospital stay of about six weeks. I have the death certificate for her premature baby, who only lived a day or so, and her own, from about three weeks later.
I went back and looked at the certificates this week, and realized that all of this ordeal happened in the same Atlanta hospital where my Mystery Baptist DNA match's birth announcement says that they were born. The timeline went something like this (I am intentionally obscuring the actual exact dates for privacy reasons):
April 1936: my relative enters the hospital
May 17: her baby is delivered
May 18: her baby dies
June 3: she dies, as well
June 11: The Atlanta Constitution runs a personal item apparently about a baby shower for the Mystery Baptist's mother, thrown by members of her church
July 27: the same newspaper runs another personal item announcing Mystery Baptist's birth on July 23.
I found one study on preterm pre-eclampsia that suggests that it is about nine times more common in the cases of multiple pregnancies than with singletons. So while some number-crunching suggests I should still be very cautious before assuming multiple births in this case, I am wondering if what happened could have been something like this:
My relative was actually carrying twins. She delivered them prematurely, and one could not be saved. The other twin was the Mystery Baptist, and against all odds, they survived despite their mother's death. But her husband, in his grief, could not cope with the idea of being a single father. In the days after her death, he opted to give the surviving child up for adoption. The hospital and adoption agency informed MB's adoptive parents, and after a brief interval to plan the particulars, their church friends threw a shower for them. A month or so later, MB was thriving well enough to be brought home, and that is when they announced the birth in the newspaper.
(Another piece that may support this adoption hypothesis is that as far as I can tell, the Mystery Baptist is an only child. Their parents married in 1930 and at least by the time of the 1940 census, it doesn't appear that they had any other children.)
A fascinating coincidence at least! Unfortunately, there are several significant facts (particularly, the details of the available DNA evidence) that cast doubt on this adoption scenario being how it really happened.