I had my car shipped up last year from FL to NJ. Many of these companies out there brokers - as you've seen. You contact one place, and they basically blast your info all over resulting in phone calls, follow ups that will come for weeks, and prices that will range from $400 to $4000 (maybe a bit facetious, but you get the idea).
I believe Reliable is who Mecum uses. They were great to deal with, but a bit pricey for me at the time. I ended up using Montway. They were easy to work with, didn't give my info out, and provided me with a couple of different quotes. Once I agreed, they gave me the actual driver and his cell phone so that I could contact him at any time. Everything went fine. Delivered a few hours early. Since then, my neighbor has used them to ship 2 cars with no issues
A few things to consider (no disrespect to any shippers here, or those that know people who ship).
You can have open transport, or enclosed. Inside costs more.
You can be top rack or bottom rack. Problem with bottom rack is that "IF" someones car is leaking above you, like break fluid, kiss your paint goodbye. Problem with top rack - if you are not in an enclosed trailer, you car can be subject to road debris. I specifically requested top rack and being the second car from the front. This way the first car took the brunt of the road debris. The cars in the back of the transport will get blow-back from the trailer.
Ask the driver to strap the car down, not chains. With chains, they will crank them tight as hell to compress suspension and give them every bit of room possible. The chains are also hooked into your frame, and over a 1000+ miles, they'll kind of "round out" the holes. Maybe not a big deal...
Pictures. Lots and lots of pictures before it ships. Get a picture of the car from every angle. Inside, too (especially if its clean). And the odometer. You don't want your car showing up a week later with an extra 5, 10, 50 miles on it.
Ask the shipper to be careful of the front end clearance. Depending on the angle, they may need to do something as simple as putting a piece of 2X4 at the edge of the ramps.
I think that's about it!