Need Help locating information in Broome County NY

I am looking for ideas on finding information on a child born between 1933 and 1943. My problem is I have no dates and do not know the childs name. The child was a boy who died at age 18 months.

He is buried in the same cemetery as my grandmother Emma Eunice Shores. I do know he is on the far side from her. The grave I believe is unmarked and in a cemetery by a Methodist Church in Vestal, NY. The church does not have a record as far as I know of him.

His parents were Curtis Rowe who sometimes used the surname Rutledge. His mother was Hazel Rowe. There is no one living I can ask about his name. He was my brother. My mother Hazel died when I was two years old.

Am am hitting a brickwall with no name or dates.