Hello and thank you for the opportunity to assist you. My name is ***** ***** I will do my very best to answer your legal questions.
First, I think that you may need to rethink who to sue. The insurance company would only be liable to you if you had some sort of contract with that insurance company, or if the insurance company itself was negligent. But if I understood your question, it sounds like another driver was negligent, and that you are planning to sue that driver's insurance company. If that is the case, then you will almost certainly lose. The insurance company is not liable for the other driver's negligence. Instead, the insurance company has a contractual obligation to defend and to pay (up to the policy limit) when the other driver is deemed to be liable. Accordingly, you must sue the other driver, and not his insurance company.
In order to begin a lawsuit, you must file in court a document that is called a Complaint. The Complaint states the facts as to why the defendant is liable, and then asks for relief (i.e., an award of money). You can see a very good sample here:
However, before you file a lawsuit, I would first talk to the would-be defendant and his insurance company to find out whether they intend to pay you what is owed to make you whole again. For example, if you were damaged to the tune of $25,000, and the defendant or insurance company is willing to give you $25,000, then suing would certainly be a waste of time and money. You wrote that you don't want a settlement, but I assume you meant that you don't want to settle for less than what is fully owed. I don't blame you for that. But it's possible that they will be willing to settle for exactly what you are fully owed. I've worked on plenty of settlements like that.
If they won't give you what you demand, then it makes sense to file a lawsuit if you believe that you have a decent case, and you can either handle it yourself in small claims court, or you can afford to retain an attorney. On that note, if you do need to retain an attorney, then that cost will need to be factored in when determining whether to settle or sue. For example, if you were damaged to the tune of $25,000, and the attorney will charge you $5000, and the defendant offers $20,000 to settle, then under those circumstances you may also wish to settle even though it is less than the full amount. In NYS, the small claims limit is $5000. So, if your claim is for less than that, then you can sue in small claims and possibly avoid having to retain an attorney.
Does that answer your question? Please let me know if you need clarification, as I am happy to continue helping you until you are satisfied.