2000 Buick Century brakes problems

Dt*: the contact stated while applying slight brake pressure at 5 MPH on a city street, the brake pedal went all the way down into the floorboard and the vehicle crashed into the rear of another vehicle without warning. The contact was wearing a seat belt, but minor injuries were sustained. Minor damage was sustained to the front of the vehicle. The vehicle was towed to a dealer who determined the brake lines were corroded. The vehicle was exposed to extreme winter weather and road salts.

2000 buick century. Frequently I apply pressure to brakes and the car will not stop for several feet. I push pedal all the way down, it makes a grinding sound and sort of vibrates but car will not stop. Several times just missed hitting other cars/people. I am also a single parent with small children in my car. I have changed brakes, rotors, etc. Several times in last couple years. Have been to many different mechanics & dealer- none can find problem. Problem most often happens when it is cold outside.