I have all of these same Fuller names and dates you mention in my line, I am descended from from Elijah II's brother John A Fuller. I would be interested in sharing information. I have Jonathon Fuller as being born in 1643 and his father's name as Robert born in Southhampton England. I do not have any information that they were on the Mayflower though.
Please contact at EwokPT@aol.com