Did Utica State Hospital really have births? You can call them and find out. I believe you would have to call the Marcy Psychiatric Center as Utica State was closed many years ago. It was a mental hospital.
Have you registered with the NYS Adoption Information/Sibling Registry? You should also register with the ISRR (International Soundex Reunion Registry). Usually, the name of the hospital where you were born is considered identifying information and not given out.
Do you have information on your birth mother? I can't tell by your post if you were an adoptee or not.