Can I still sue my former NY employer for an injury I suffered 3-3.5 years ago?

Unfortunately you do not have an ability to sue for two reasons. First you can't sue your own employer. You made a claim under Worker's Compensation and that is all you are entitled to as far as your employer is concerned. There are some very specific and limited exceptions to this which allow for an employee to sue his or her employer but this would not be one of them. Secondly the Statute of Limitations for this type of claim is three years from the date of occurrence. As your accident happened over three years ago your are barred from bringing a lawsuit even if there were a third party besides your employer who was responsible for the happening of your fall (say a cleaning company hired to keep the premises free of snow and ice).

I dont practice in New York, but after so much time, you almost always have to worry about a statute of limitations issue. I would speak to an attorney ASAP and see if the statute of limitations has run first and then see if you have any exceptions.

My standard disclaimer: I am not offering legal advice, assume I do not know the law in your state and that I am just making suggestions for starting points for when you do speak with an attorney. Do NOT rely on anything I write and contact a lawyer in your area immediately after reading my posting.