I had a hell of a time getting a Yukon shipped from Massachusetts to Washington state. I wanted it covered and in vehicle shipping they use the term "lanes" a lot. The problem I had was starting way up north, cross country, then north again...I waited two months.
As others have said you need to be aware most shipping companies broker out your transportation. There are some very good independent drivers out there and others who will mess up your vehicle, you never know what you will get with a brokered load.
As far as national carriers go some of the best rated include Reliable and Intercity Lines, they are very reputable. If I was to go brokered I would look into the Bring a Trailer auction site of classic cars, they offer a shipping option. BaT will arrange shipping like a broker but are very picky on the carriers they use and will boot those that don't perform off the list. Check out bringatrailer.com for their shipping options.
The carriers are all slammed, the Bring a Trailer option might be the quickest with likely good results.