I don't know if you are still following this or not, but Panina of the 1850 household was Panina Hamilton, a 35-year old spinster with an illegitimate son whom John Morgan married in Franklin County, Georgia on 30 July 1842. They did not have children together. The Alonso F. "Morgan" shown in their 1850 household was enumerated as A. F. "Hamilton" in the 1860 census. He was almost certainly the illegitimate son of Panina, as a glance at the 1840 Franklin County household of her widowed mother suggests.
Depositions recorded in Lumpkin and Cobb counties concerning claims of back pension payments to her mother make it clear that Panina Hamilton was unmarried prior to 1842 -- she was the second-born of four children of Joseph Hamilton and Mary Hall Turk. The depositions concerned a share of her grandmother's Revolutionary War widow's pension which was collected by her two surviving children William Turk and Mary Hall Hamilton. Mary Hall Hamilton died in Walker County in 1853, according to that testimony, and her four children split the money due to her. Panina Morgan's younger brother John T. Hamilton was a neighbor of the Morgans in Walker County. Her other two siblings lived in Franklin and Cobb counties..
Panina Morgan was alive as late as 1853 when she claimed her mother's pension payment, but I did not find her anywhere in the 1860 census. I would note that her full name may have been "Mary Panina", as her son Alonso Hamilton and stepson James Morgan each named a daughter Mary Panina and a few of the Morgan children named daughters "Panina" as well. Although she didn't marry John Morgan until 1842 she must have been influential within the family.
As you know, the courthouse fires of 1883 destroyed what hopes we may have had of finding probate records for either John Morgan or Panina.
I am descended from John Morgan's son James via his daughter Margaret. I haven't been interested in this family until recently, but am stuck on John Morgan's ancestry. I have yet to find convincing evidence to identify his parents or siblings.