I have shipped three cars in the last year (several others in the past) None of the cars this year were anything that special. Older Chevy Van, Non-running Cayenne and 914 roller.
Most of the general auto shipping companies are just a shipping broker. They put out a request for transport from A to B and see who has space and what the cost will be. They will add their fee for this and then get you a quote. These will generally be the lowest cost way to ship. I have had everything from 18 wheeler loaded with cars to guy with a pickup and trailer show up. I would not use this method if I was shipping a really nice or unique vehicle.
The other shipper are what I would call hi end shippers. These are the guys with their own enclosed trucks. Think Reliable, Intercity, Horseless carriage etc. I have not used any of these.
If you search for Auto shipping you will find plenty of options and can look for one with good reviews. Also be aware that some of those sites offering to find you shipping cheap will result in being bombarded with emails and texts as they just throw your information out to the word of shippers.
I used a company called Viceroy for the last two shipments, they are just a broker, and was happy with the experience and would use again. They will also ship non-op vehicles which many will not.
https://www.viceroyautotransport.com/john