I think I can pretty much answer everything for you. First, the resource you are looking for is Transport Reviews.com ( http://www.transportreviews.com/). It is a ratings and review board with over 50,000 reviews on over 1,500 car shipping companies.
So there are lots of pointers on there. Unfortunately I have to tell you that you are not allowed to put items in the trunk of the car. The license to ship vehicles does not allow them to ship cargo or household goods.
You can still find companies that will do it though, although you will be told that they are not responsible for the items. So if you find a company that doesn't follow the license standard you want to make sure not to have anything of value in the car. Along that line shipping books in the car could create other major headaches.
Trucks are limited in the weight they can carry. So when they build a truck load they are counting on the weight of the vehicle, without the weight of books and clothes. It may only be 100 or 200 pounds, but if the truck gets stopped at a weight station for being over weight you can count on your car being left behind for someone else to pick up.
So if at all possible I would try to make other arrangements. You sound like you have an idea of the cost, but I'll summarize some of what others have paid. From California to Florida the reviewers have most commonly paid $850.
The average cost to ship is $1,002. For price quotes for your specific move, from companies rated well on Transport Reviews you can submit a quote on AutoTransporter.com ( http://www.autotransporter.com/). Hope this helps.
Good luck.
Considering the expense, it would be cheaper to drive and fix it up along the way. Last time I checked with some one who was licensed and bonded for such tasks they charged about 3.00 a mile (that means 9000 dollars about). You can rent a u-haul truck and trailer for that much money and tow it yourself.
If you are on a real tight budget, it might prove best to sell the one you have and just buy a replacement when you get to Florida.
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