Car accident with no insurance?

I was stopped at a red light behind two cars. Both cars started to go and so did I the next thing I knew I had hit the car in front of me. The car in front of me hit the car in front of him. We all pulled over my car isn’t to bad but I drive a suburban, the car in front of me was a smaller car so his looked a little worse. Nobody was hurt, and all cars were drivable, so we decided to not call the police we just all exchanged information. After we had finished and they left I immediately called my insurance lady to let her know what happened. I am not super clear on this next part, we have been going back and forth with our insurance since that call and will continue to fight it. When I called the insurance lady she said I wasn’t covered. She said that they had received the last payment much later than it was due (not sure how because it was mailed the same time as always) but because of that our insurance had been cancelled. A couple of weeks ago they had chased our check, she said we were reinstated as of yesterday (the day of the accident) but because it was the same day they weren’t liable to pay the damages. We are going to keep fighting them because I don’t think its fair. Even if they decide that they won’t pay we can afford to just pay for the damages to the rear end of the car in front of us, because know matter what happened I am at fault for following to close. We can’t afford to pay out of our pocket also for his front end and the rear end of the first car. Shouldn’t those damages be covered by the car I hit because he was obviously following to close to the car in front of him? I’m a little freaked out with all this, this is the first accident I have ever been in and we have always had insurance, so we thought. The insurance company did say they mailed us a cancellation notice but if they did we have never received one.