I have gotten many birth, death, and marriage certificates from the NY State Dept of Health - vital records. Requests that are sent in by mail take a verrrry long time to complete (a couple of years ago the wait time was about a full year).
Since I live near Albany I just drop off my requests and have gotten the certificates I requested in weeks (2 - 4 weeks).
One issue that might be getting in the way of success for you is that number you see on the copy you currently have. Because vital records are registered with both the locality/town/county and with the state - your record might have the county/locality number on it, not the state number (which would be a different number) - depending on the source of the copy you have. If you are asking the state to find your record using a local number - they would correctly tell you it does not exist. Unfortunately NYS will look for exactly what you tell them to look for - and nothing else.
You need to access the nys vital record index to get the correct state number, or pay them to do a search for you, then request the certificate from the state number.
contact me directly if you need more info.