If I was driving to Alaska I’d rent a car so I would know it was fairly new and well maintained. But that’s not the only way to get to Alaska. A person could drive a rust bucket to Alaska too, and likely get there, perhaps not in the same car they started with, and possibly not at their estimated arrival time. But they’d probably get there. Everyone has their own methods, and their own schedule.
Flying to Alaska works well. If your BF still has some of your belongings he can ship them UPS. Furniture and appliances (I don’t think those would be transported in a Corolla) can be sold or given away.
There are different ways to get to Alaska. If you don’t like driving the distance you can get on the Ferry in Washington state and get off in Anchorage. It’s a very scenic trip, but costs more. The Ferry takes all manner of vehicles.
The Alaska ferry system is a great way to travel. To get to Anchorage one gets off in Whittier, a short scenic drive from Anchorage:
I agree this is probably a lost cause. The only thing you MIGHT consider is pulling the engine and transmission if doing so would bring you considerably more than the tow fee. Getting the tow fee paid free and clear might just be the better choice if you ask me since you don’t have to worry about the hassle of selling something quite large and hard to handle. I am sure there is someone who would love that engine and transmission but finding them would be the hard part.