Your suit is against the other owner and/or operator. The insurer only steps in to indemnify and defend the insured. You should consult a personal injury attorney if you really believe that there is a basis to sue, although you advertise that you illegally ran a red light. Liability of the other party, not just causing you injury, is a necessary component to an injury lawsuit to be viable and successful.
It sounds as if the officer "overcharged" you. There is case law that may make the going over the double yellow not prosectuable or at least questionable, and there is not enough there for reckless driving in my opinion. The DWI strictly depends upon our reading. Since you have CDL, you should fight this all the way. Hire an aggressive and experienced DWI attorney to combat this irate police officer.
Under no circumstance, unless you are a lawyer yourself or similar exceptional situation, should you fail to have a matrimonial attorney review your agreement for accuracy, completeness, and advice as to the hidden effects of language used or omitted. Yes, it's an investment, but saving money now could result in significant losses/liability later. If money is that big an issue, contact the local bar association for mat attorneys who will work for a reduced fee.
It would take a treatise to address all your rights. Mr. Palumbo makes a good case, but most people won't do all that. The absolute best thing is to not drink and drive. It only gets more punitive every year.
It stays on your computer record as long as there will be a DMV data base. As you get older and stay arrest free, it will be easier to argue that it was a youthful error, and not a potential pattern to employers. Legally, there's nothing that seems helpful.
The traffic ticket will be amended. Your best bet is to have your or your lawyer discuss this with the municpal prosecutor in Beacon to bargain the disposition, unless you were not speeding and wish to take it to trial.
A private attorney, experienced in DWI litigation, may be able to afford you more qualitative time in representing you. But how are you going to afford one as one must be found indigent to obtain the services of a public defender?
New York has a statute of limitations of 3 Years from Diagnosis, 2 Years from Death for Mesothelioma claims. Certain calendars in large municpalities, such as NYC, are devoted to such cases. You should contact counsel experienced with such claims to determine 1) if you are still within the statutory time period to sue and 2) if you have a decent claim, which it sounds that you very well may.