Should we try and search for any birth records in Tennessee for Francis/Franklin/F M Davis?
That would be a good idea. When you do, also search for his father. Look in the 1850 census for any male named Davis, born say 1770 to 1810, who was living in Tennessee but was born in North Carolina. That allows for his father being anywhere between 20 and 60 when Francis was born. If there is a son named Francis in the household about the right age, make a note of the family. See if their Francis was still in the household in 1860. If so, they aren't the right family, since we know your Francis was on his own by then with a son William - whom you are sure of.
If Francis is not there, then research the family like it was your own and see if it lines up. Were they in NC in 1840 and before? Where? near the Colliers by any chance? A negative answer to that question wouldn't rule them out, but a positive answer might make them a little more likely. Did the parents end up in Stoddard County like your Francis did? etc.
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I found a family in Kentucky using that method but never worked any more on them. You could follow them up also.
They are:
1850 Hopkins County, Kentucky
William M Davis, 42, farmer, real estate: $800
Elizabeth Davis, 45
Frances Davis, 18, male, born in NC
Emisetta Davis, 14
Benjamin Davis, 10
Israel Davis, 8
Mason Davis, 6
all born in NC
https://www.familysearch.org/ark:/61903/1:1:M654-XQJ?lang=enThat Francis is not with the family in 1860
https://www.familysearch.org/ark:/61903/1:1:MZBQ-5B2?lang=enThe fact that their Francis was said to be born in NC, and probably was since all the younger children are said to be born there also, does not eliminate him in my opinion. At this point I don't think we really know where he was born.
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Here is another person I never got back to look at:
1850, Buncombe County, NC
Francis M Davis, 18
https://www.familysearch.org/ark:/61903/1:1:M4BW-3K7?lang=en