Lincava:
You have a three year statute of limitations in New York (N.Y. Civ. Prac. section 214).
You have a serious injury with serious complications. Ask the insurance company for the adverse party to disclose their policy limits. If the limits are minimal, and the adverse party is otherwise judgment proof, then demand the policy limits. Obtain an affidavit from the insurance company and their insured that there are no other collectible assets or insurance coverages. This is not difficult and will not require a lawyer to verify. Consult with a lawyer (who you can pay by the hour) to prepare and review any releases or affidavits required. It is not necessary to hire and pay a lawyer a percentage fee of a minimal policy limit that the insurance company will probably pay anyway.
If the adverse party's insurance limits are inadequate to cover your claim, then you also can make an underinsured motorist claim under your own auto policy if you have such coverage. This claim also does not require a lawyer if your claim is worth more than the policy limits which you maintain.
If the adverse party had high liability limits, then retain a lawyer on an hourly basis to evaluate your claim. You can then make the claim on your own. Of course, if you can't get the insurance company to offer what the claim is worth, then and only then should you hire a lawyer on a percentage fee basis to file suit and take your case to a jury. Even then, you should negotiate with the lawyer to only pay a percentage of the amount recovered above and beyond the amount offered to you before you hired the lawyer.
Don't forget that you also should make a claim to your own auto insurance company for PIP or medical payments benefits for your medical bills incurred.
Hope this information is of some help to you. Good luck, and don't give up. Hope you are feeling better. Be sure to follow through with your medical treatment.