If you ask me I'd say sell the car down south and don't worry about shipping it. We ran into a similar situation when we moved to ANC, the military would only pay to ship one vehicle. We ended up selling one of ours and buying another one up there used. There are plenty of options in ANC for buying cars, only the brand-new ones off the dealers lot will pay the markup for shipping to AK. It was about $2K when we were there, versus the $700 or so you'd usually pay in the rest of the US. BTW, lots of cars up there are equipped with a remote start system, a very desirable accessory if you don't have a heated garage.
If you drive it to Tacoma (I think the actual terminal is in Bremerton) you'll find that the shipping company won't guarantee that the belongings inside will make it to Alaska...they just ship cars, not stuff. My wife had some skiis and such in the back of her station wagon when she showed up at the terminal and they gave her a hard time about it but finally just let her put in on the boat with a signed waiver basically saying "if all your stuff dissappears don't come to us, we told you so). Its pretty much that way wherever you ship a vehicle unless you are going along with it like on the ferry.
If they're bound and determined to take it up north, you might consider the Alaska Marine Highway System.
http://www.akferry.org/ Its the state-run ferry, and carries cars, trucks, trailers, RV's, etc. including the passengers. You can't carry stuff like extra gas or propane tanks, but regular non-hazmat belongings are fine. The cabins are pretty much sold out for the summer but most people just bring along an air mattress and sleep in the solarium or on the deck chairs indoors. They have showers, lockers, a cafeteria, etc. Takes about three to five days or so from WA state (Bellingham) up to AK, depending on your destination and the boat. Price depends on the length of the car and the length of the trip.
Unless the car is somthing special, like a 4wd or some sort of sports model you'd probably get a better price and have better luck selling it in the lower 48.