not insured, we're hit at stop sign by insured driver. do we have any rights?

[fenderbender]

What is the name of your state?NY state. My wife was sitting at a stop light with our 3 children in the back seat. Teenager hits the car. We exchange info, though choose to not go thru insurance, as it is only a fender bender, (plus, our insurance has lapsed, though it would go under hers).
They have not called us, nor returned any calls. Finally I went to their address, parents answer for her, apologize for not getting back. Then they take the estimate I got on the car, (very fair, legit, printed up), & they want their own guy in a town 45 min away to look.
I am upset that they never got back, were trying to get away with doing nothing, as it's been over 3 weeks, & they not once asked if the 3 kids under 10 were ok. Instead, their only concern was that $500 was too much for a rear bumper.
I would love to make them go through their insurance, where their rates would go up with teenage driver, plus I'd get a rental if my car was repaired. But I would rather collect the money, which I could use to get insurance again.
Do I have any rights, or have I forfeited them having an uninsured car in the road?
Is it worth it to face whatever fine/penalty in NY for driving without insurnance, if it's less that I would collect from them?
And, although the kids were sore the next day, back & neck, we never followed up with a doctor, & they seem ok. Shouldn't they be concerned about us filing if the kids got hurt? My youngest, under 4, cried when I got to the scene & tried to put him back the in carseat. Was scared to get back in for a bit.
Again, now he seems ok in car, but do I have any rights?

 

[Dougthegreat]

Again, now he seems ok in car, but do I have any rights?

Yes. You have the right to keep insurance on your vehicle, to protect yourself financially in the case of a collision. Some might even say you have the obligation to do it, seeing as driving itself is a privilege, not a right, and comes with responsibilities.

Here is a list of your other rights:
Here

 

[MandyD]

Your lack of insurance means nothing when determining fault in an accident. The penalty for driving without insurance in NY is threefold. Your wife can expect to have her license suspended and probably a court imposed fine. Whoever the car is registered to has already begun to accrue fines of $4 a day for everyday that the plates weren't surrendered after the lapse of insurance. The moment your insurance was canceled, the DMV was notified by your insurance carrier.

 

[fenderbender]

rights

Thanks. So, I know I hvae the right to insure my vehicle. Yes, driving is a privilege. But now that I have been hit by a driver who wants to avoid having THEIR insurance rates go up, as a teen driver at fault in NY drives it up big time.
But do I shy away from making sure these guys pay for their involvement in this accident? She admitted it was her fault. But now she is taking her time in dealing with it. I hope it's not because she thinks she can get away with it since I was not insured at the time.

 

[stephenk]

so take the other driver to small claims court for the $500 or you make the claim against her carrier.

In exchange your wife will lose her license for driving without insurance and will be fined by the DMV for not having insurance. Is that worth arguing over $500 or something close to that amount? You can always agree to a lesser amount.