Just shipped my '49 in February from Minnesota to New York.
I was referred to this site:
http://transportreviews.com/ to start off with and read about the shippers. It was my first time shipping so I wanted to understand how it worked.
You call one of these shippers and they act as your broker (I went with John Andrews from Envision Auto Relocations). You can give them a price range of what your looking to work around (the gentleman I bought it from said I could keep it there a month if I needed, so I had time on my side) or you can just lock in one of their quotes (my range I wanted to work with was between $900 - $1100 including their $150 broker fee, these broker fees vary so make sure you ask). My quotes ranged from $600 - $1140. One of my father's friends ships cars all the time (big dollar Hemi cars) and he had said "you don't want to go with the cheapest and you don't want to go with the one that is the most expensive, go in the middle" (this later was confirmed by another broker who said the same exact thing).
So when I closed on $920 for shipping they provided me with the name of the shipper and I did my research on them. Again once they give you a quote, you can get the name and do your homework on the shipper. They were an absolute solid reputable transporter (Bill's Transports) and I would HIGHLY recommend doing business with them.
My truck did not run, the brakes did not work, and it was no where close to the St. Paul - it was in the middle of no where so I was extra charged for all of this. So they required a truck with a winch. Bill's Transports picked up the truck the next week after I received the quote and it was delivered in 3 days. He called me when he picked it up, when he was half way to my location, and 5 hours before he dropped it off. The communication was key to me and they were ranked high for that.
Again just make sure they are fully insured and have a number of positive reviews. Good luck.
Any questions PM me and good luck.