We had our Leaf shipped using Plaza trucking. The route is really more important than the company use choose as they are almost all brokers for a network of trucks that handle certain routes. If you can swing a good deal with the broker (average fee is $200, we paid less) and can get a decent quote of the price (we paid $550 for the ride from Tempe, AZ to Tulsa, OK) that's about what you can expect. It was BullDog trucking that finally picked up and delivered our car (in record time, we contracted the ride on Monday, was picked up Tuesday and arrived on Friday). It really just depends upon the route and driver timing so YMMV.
This was for uncovered (seriously, it's a car and it likely came to Nissan uncovered -- yet wrapped) transport. The driver has never driven an EV (or a Leaf) before but the dealer gave him a quick introduction and all arrived fine. Uncovered transport is more than fine, I mean, it's not a Ferrari or a ZO6 vette, but everyone has their ideals on car safety I suppose.
The car arrived in fine shape, there was some dirt/grease on the driver inside door but it came off with a quick wipe of the interior (the white glove treatment doesn't matter when you wear the gloves while operating the lifts). Your experience may vary, of course, depending on the driver.
Since driving the car was not an option, and renting a truck/trailer to make the 2000 mile round trip journey is just not feasible nor cost effective, it's really the best way to ship a car (ours was an orphan) cross country. There is a site when you can post your route and information and get calls from companies that can do, I think we heard back from 10+ companies in about an hour, but I can't remember the name of it. I can get it from my wife later.
PS: make sure dealer fully charges your Leaf before it gets on the truck. The driver will likely not know how to charge it and will need to move your car as other vehicles are add/removed from the truck during transport.