Welcome to Just Answers! Thank you for giving me the opportunity to assist you! I will do my best to help!
I hate to be the bearer of bad news, but the tax warrants are issued by New York's Dept of Tax and Finance's Collections & Civil Enforcement division. They are only issued after the period of time for appeals has expired. A tax warrant is the equivalent of a judgment against you. About the only recourse that is left to you is to try to make an offer and compromise.
There should have been several opportunities for you to appeal the decisions and to request hearings or request Conciliation Conferences, which are held by the Bureau of Conciliation and Mediation Services. Your specific rights and the timetable that should have been followed are found NY Publication 131, Your Rights and Obligations Under the Tax Law. You can access it here: http://www.tax.ny.gov/pdf/publications/general/pub131.pdf
If you can prove that the State missed a step, you may be able to file a civil suit, or perhaps simply bring it to the attention of the person who was supposed to hear the appeal/meeting and did not. But this is a long shot. They usually make sure that they have followed all steps in order to avoid having warrants overturned.
Similar to the IRS, the State can find you a "responsible party" if you had the authority to issue checks or approve payments and did not. You do not have to be the majority shareholder or even a high-ranking officer. If they find that your position in the company gave you the authority to approve checks for payment, you can be found responsible.
If you do not have the $80k, as you stated, and you still have issues outstanding with the Feds and the other states, you may be in a position to get an installment agreement, and perhaps even an offer in compromise, where you settle for a fraction of what you owe.
I wish I had a better answer, but the law is the law! If you have any more questions, please let me know. I hope you have found my answer helpful! If you have, I would appreciate it if you could rate me highly!
Again, thanks for your question! Have a great weekend!