Here's one place to start, and there are others. The two basic methods are containerized and RORO (roll-on, roll-off). Assuming you can get hooked up with a RORO schedule (should be easy to get space as I thing they would be more full coming from Europe than going to), I think that could be cheaper. Not sure, but I think it could be if you are in a position to deliver/pick up at each port.

So, what car?

Several weeks ago in Tarbes, France I saw a *completely* unremarkable ~1969 Dodge Dart on a trailer being towed by something like a small Opel. What, indeed, was up with that??

10 days, IMO, is enough for Florida, unless you feel a compulsion to stuff like Disneyworld. The drive can be a pain, but consider taking in the Florida Keys. Like to fish??