I am a long-time Fleming researcher just "passing thru" this forum, but thought I might give you a little info. According to my info, Elizabeth M. Adrian married William McDonald ca 1813 in Franklin/Banks Co., Ga. They had 8 children.
From the McCall Collection, Georgia DAR, "Elizabeth McDonald, widow of Wm. McDonald, applied for bounty land Mar 22, 1855, for the services of her husband, William McDonald as a Soldier in the War of 1812. Was granted 160 acres. In 1871, she was residing in Murray Co., Ga., and in 1874 at Spring Place, Murray Co., Ga. Ref. from Pension (including Bible) Record on file Veteran's Administration Building, Washington D.C." William and Elizabeth McDonald are buried at Spring Place, Murray Co., Ga.
Elizabeth M. Adrian had a sister Rachel, and may have had another sister Mary. We Fleming researchers believe it was one of these other sisters who married John Davidson. I have no info on children. And no other known source that definitely names John Davidson's wife. Good luck....... Lisa Sheppard