Older LRP Runners, they go forward, brake then reverse, but its just as quick as you move the stick, you wont even notice it brakes, it will revese really quick.

Not to insult your project. But it sounds same steering design as Tyco and Nikko rc. They are non proportional, and the motor just keeps pushing the sterring rods when you hold the stick on either direction. Also you will need some kind of spring to hold and bring back the wheels in the straight forward position.

Have you considered or tried using a High Speed Servo?

Can you share photos of your project. Someone here might be able to assist you more with the help of some visual aid.