I haven't personally used any car transport companies before, but I've heard good things about
open carrier car transport. It's a pretty common way to move cars across long distances. You might want to do some research online and look for reviews from other customers to find a reputable company. Make sure to get quotes from a few different companies and compare their prices and services.
Also, I'm not in the Melbourne, Florida area, but you might want to try posting in a local Facebook group or Craigslist to see if anyone has any recommendations for transport companies. Sometimes it's helpful to get recommendations from people in the same area.