if you google 'auto shipping' you can find listings for sites where you enter your info once and get quotes from multiple companies. I did this when I shipped a Miata from Miami to San Fran. I was not hounded in the future ... got quotes and contact info and that was pretty much it. Selected a company and called them.
Note that this service is typically not door to door .... in the same way that household goods can have multiple homes on one truck and it makes additional stops so to must an auto carrier fill his truck then proceed with potential stops along the way. In my case on the selected pickup date a simple flat bed tow truck came and pickup the car to take it to a holding location. I did not get the car back at the end point for several weeks where it came off a regular big auto carrier ... obviously not the truck that picked it up.
As mentioned it is not cheap - and this is for an open carrier. If the car is 'special' and/or you want a closed carrier, this costs more.
did you know you can hire a driver to drive your car? This can cost much less .....
https://guides.uship.com/autos/hire-someone-to-drive-car-to-destination/
also while it may not help today, something to remember ... I was looking to get a car from Florida to Norfolk and didn't want to drive so was again looking at shipping costs. Mrs suggested the auto-train. I put the car on the train and got a roomette for a fraction of the cost of trucking the car. I had a lil bit of a drive on each end but under 2 hours .... I've used this method to move cars between VA and FL a few times now.