I'm a NYS Criminal lawyer.
You can get a certificate of relief from civil disabilites, by applying to the Department of Parole. Here is the application form
which will explain just what it will and won't do. But basically, the Certificate will not erase your conviction(s) but if granted it will restore your civil rights.
The Certificate of Good Conduct will restore your right to sit on a jury, to run for office AND -- but only if explicitly granted when you get it back -- your gun rights. The Certificate of relief will not restore your right to run for office and will not give you back full gun rights. Make sure you know which of the two certificates you really want before you apply.
You don't need a lawyer to fill out the form for you.
I am not sure why you are interested in Correction Law § 753. Basically, if you haven't yet received a Certificate of Relief and been denied employment because of your criminal record, it doesn't apply yet. The section just defines when a NYS employer is allowed to deny you a job because of your record, even with the certificate.
If you are asking whether or not it ever gets too late to apply for the Certificate, the answer is no. In fact, the longer the period of time and the longer you have stayed out of difficulty with the law, the better a candidate for the certificate that you are.
If you're looking to sue an employer, in general, it would be 3 years. If there was an employment contract you'd have 6.
I think I have answered your question but use the reply tab below if you have anything further.