What kind of lawyer do i need for a auto accident when the Lawyer gets Paid if a settlement is made?

I'm currently active Duty army and i was informed that their are lawyers that work traffic collision claims and don't get paid unless their is a settlement and I was wondering if that was true. I was injured in the accident last august I still have the police report and I also have my medical documents. Due to the accident I have 2 buldging disc in my back L4 and L5 that's penetrating me root nerve system. I'm now going thru the process of being medically discharged from the army and my career progression and chance of being promoted and staying in are slim to none. Orthopedics is telling me that I am no longer fit for duty and able to perform my military duties since I can't stand but for a period of time cant lift, run, jump, wear my battle gear and more. I want contacted by his insurance company about the incident or a possible settlement. I want to see if i can start a settle/law suit for the compensation of force of discharge and pain and suffering.