Kelly,
I called the Onondaga Medical Association and they said the birth records from St. Mary's are kept at CC. I talk with a Sharon at CC in Syracuse, NY about my birth records from St. Mary's. She took my info and said she would get back to me. Her number is:315-424-1800 ext 305. I just talked to her on Friday so I haven't heard anything. Call her and see what she can do. She seemed wiling to help.
Good Luck.
Kathy