I just went through this, shipping a 66 Charger from Dallas, TX to Rhode Island for $750.00. The broker I worked with gave me all the details about how this works. Keep in mind that when you are looking at brokers (ie any company in Florida), they are only the middle men, so it does not matter who you go through unless they personally own their own trucks, in which case you will pay more.
When you contract with a broker, they post your route on a master board that ALL of them use, and the drivers simply enter their current route and all the available pickups will show up. The going rate a couple months ago was $0.35 to $0.40 a mile, so get the distance you are looking for, multiply by $0.40, and that should give you a high estimate for what it should cost. It's basically like an auction, the more you offer, the more likely it will be picked up faster. When estimating your price, keep in mind that the broker will keep $150 or so as their fee, and the balance is what goes in to your cost per mile.
Based on your zip codes, you should be right in the ballpark of what I paid. If this is not a show car, I would not spring for the enclosed trailer IMHO.
P.S. To answer your question, here is a good place to find reputable shippers:
http://www.transportreviews.com/
