I shipped 2 cars cross country on my own dime. I first started with some website I found using google. It was basically a job board with people competing with each other. I'd get a new e-mail from the board system every 15 minutes as drivers were outbidding each other by $2.
It was my job to pick the right driver, based on price, insurance, reviews, etc. I got scared and asked some RL friends. I ended up getting a referral to a transportation broker. It ended up being about the same, but I felt a lot more comfortable having someone with experience pick for me. I guess it actually cost me more money because I had to pay a (?$50?) broker fee. But I was happy to do so because my friend had shipped his car back and forth across country using the broker both times, and no problems with the car.