Like you,it seems hard to imagine a Francis Dossett and a Francis Dorset living in the same county at the same time.However, I have been unable to find any descendant of Francis Dorsett who used the surname Dossett.
I do not have all of my notes in front of me.Orange County, North Carolina was created about 1755.I have not yet searched the records of the county from which Orange County was created for records of our ancestors.My ancestor Moses Dossett was living on the western side of Orange County.Francis Dorsett was on the northerly side, which became Chatham County in 1772.Francis is listed in the index for the 1800 and 1810 U.S. Census for Chatham County as having the last name of Dosset/Dossett.Solomon appears in the 1790 Census as Dossitt, 1800 as Dosset and 1810 as Dossett.Clement is listed in the 1790 Census, Chatham County, ad Clemment Dosit. A Clem Doset was enumerated in Burke County, N.C. in 1810.A Henry Dossett was listedin Chatham County in 1810.John Dossit in Chatham County in 1810 and 1820.But I have not found any record (yet) in which any of these people themselves used the surname Dossett.
I have not reviewed the Will, but the index to Wills indicates the Francis who died in 1813 used the surname Dorset.Similarly, the Will of Solomon Dorsett was probated in Chatham County in 1849 (I have not reviewed that Will, yet).There is a set of books coplied by a person having the last name of Saunders, entitled COLONIAL RECORDS OF NORTH CAROLINA.Government records are transcribed in the set.Using this set, for some reason, I had decided that the Francis who died in 1813 was born about 1740-but I do not know why.In any event, Francis Dorset was a member of a group called the Regulators (the government labeled the group insurgents).The indictment and most other records involving the event use the surname Dorset.However, the order dismissing the indictment uses the surname Dossett.
Hopefully when you go to North Carolina you will be able to find the records that tell us where the Moses and Philip Dossett, who were living in Orange County at the same time as Francis Dorset, came from, who their parents were, etc.