HI:
I would put both men down in my notes as possibly being the father, but I would still feel leery of deciding that either one was definitely the father, because Census records aren't the most reliable either. I've seen relatives' birth states change. One of my great-uncles was born in OH but he told the 1920 enumerator that he was born in VA (like his wife).

Have you found marriage records for her parents, either the mother married to George Britt or to Tom Adkisson/Elkins?

Have you found Sallie's marriage record?
Sometimes marriage records name the parents.
Question: you have several surnames for Sallie: Britt, Adkisson, Taylor, Hall, it can't be all of them, so are you sure it's the right Sallie living in Carroll Co, TN in 1880 and 1870, and not another woman with the same first name? Just askin'....

I would go to the source of the death certificate and look at it myself. I did just that with my great-great-grandmother. Couldn't find her death certificate for anything till I went to the archives -- her name was misspelled in the index. (Nothing can stop you from looking at the record.) Yes that is time-consuming & expensive but may be the only way to be sure.

Finally, a suggestion that you search on www.familysearch.org. Go to the Search Records tab, then click on Record Search Pilot. You can type in her name and also search using the parents, state, county, spouse, etc. as the filter. It's a terrific site, free to use.