States have the right to waive sovereign immunity (the presumption that you can't sue the state civilly), however, each state can set it's own terms under which they can be sued. In Arizona, the Notice of Claim deadline is 180 days after the plaintiff knew of the injury. I don't know New York law, but a quick search says that the deadline is 90 days. There are some reasons why that deadline may be extended, but if an experienced attorney told you that none apply to your case, then filing a lawsuit would be a waste of time and money since it would likely be dismissed for deficient notice.
There are ways around the 90 day notice requirement. The easiest is filing a lawsuit under federal law, speicifically, 42 USC Section 1983. This can be done without prior notice. You should contact other attorneys in the upstate area who handle such cases. Good luck.