Solenoids in RC control car

Neat - so the solenoids are connected together and need a push-pull drive?

I guess the easiest solution would be to use a 4 channel H bridge controller. You can get several types on eBay or an L293 chip could work well. That would allow you to control the motor and the steering.

Another option is to cut the controller board in half and try and use that, but presumably one of the transistors has blown if it will only steer in one direction.