I am also a descendant of Esther (Fuller) and Eleazer Kellogg, through their son Elisha who married Persis Dunham.
I found this:
"Abijah Fuller and Esther/Hester Arnold married August the 7th, 1746, in the 20th year of his age... The church records at West Barnstable show that Abijah Fuller returned to Barnstable with his young wife, and that they lived there until May 2, 1749, when they again returned to Middle Haddam, Conn., this time to remain." - The New York genealogical and biological record, Vol. 82, page 5
Can we confirm whether this is 'our' Abijah? If so, Esther was named for her mother.