Thank you for sharing your family information. I'm a Fulton hunter and would not be surprised to learn that Robert Fulton Neely, b 1799 PA, is the son of a Fulton female, b bef 1783. Yes, there were several Fultons in early Jefferson Co, OH.
As for Summit Co, OH, I found evidence of a Joseph Fulton (b 1790). There are three stones at the Bath Cemetery:
(1) Margaret Fulton, d 1 July 1863, age 76 (b1787), "wife of J. Fulton." (2) In memory of Joseph Fulton, d 1845, age 55 (b 1790). (3) Frank, son of J/I? and E.A. Fulton, d 1848, age 6m 6d.
I suspect that the "memory" stone was placed when Margaret Fulton was buried in 1863. But the stone for infant Frank Fulton was probably placed in 1848 suggesting a Fulton family in the immediate area at that time. My guess: Joseph Fulton died in Summit Co, OH, and infant Frank Fulton, a probable grandson, died here, too. Not sure about widow Margaret (-?-) Fulton. She may have died elsewhere and brought back to Summit Co. to be buried near Joseph and Frank. (I do not have easy access to the 1850 US Census of Summit Co, OH.)