There are a lot of good boats out there in your price range but I would think that it would be next to impossible to live year round on a small sailboat in Alaska. I can't imagine that you can store a boat in the water in winter in Alaska and so many of the important life supporting systems will not operate. But more to the point, even a log cabin has more insulation value than a fiberglass boat, making whatever heat source that you have go much further in terms of making life liveable. Small boats without cored hulls have almost no insulation value at all, and so are very hard to keep comfortable.
I don't know if it is impossible, but certainly it will not be easy. See if some locals do it and talk with them. They will certainly have a lot of knowledge about the local conditions.
I believe that you will need a solid boat with a small interior space and with a good heater.
Something like this:
1984 Vancouver Sail Boat For Sale - www.yachtworld.com