Shipping a car across country?

Get AAA and drive it. You're going to pay for someone fuel either way, which theirs is likely diesel (more $$$/gal) and possibly marked up, plus someone else's trip to get it. Get AAA Premier (if you don't already have it, get it no matter what), and look at Amtrak too. I can take an Amtrak across Missouri in 5.5 hours, it's a 4-4.5 hour drive, and Amtrak is $36 round trip. Sleep, eat, drink when I want. It'd cost me a tank of gas one way to do that at $40+
I paid $660 for open transport on a pickup from Denver to Missouri (just across Kansas line) in 2013 when fuel was cheap. He didn't hurt my truck, but it was mildly stressful. For your trip, take some basic tools and parts, but you'd potentially be in the upper northeast in the beautiful fall, through to Phoenix before it snows.

By my math, you're looking at about $600 in gas at 18mpg, and two to three overnights at $80/each for a decent place or $40 for Motel 6. Meals are the same whether you eat 'em at home or on the road. The wife and I would just buy a cooler and sandwich makin's and the higher end hotels have breakfast (which you can take some with you in the cooler:p )

So $840 and add a hundred bucks for incidentals. Compare that to shipping it.

And you'd likely pass through Kansas City, where I might just buy you BBQ.

Obviously, this depends on the car. Get the production dates off the tires, get pics of the tread wear and all that.