Gee, I'd say just pull the tank. It isn't that big a job, and it's going to need to be done one way or the other.
These tanks are formed from two pieces of sheet metal pressed into "bowls", and welded together. In wet climates (does it ever rain there in Seattle?) moisture accumulates in the foam between the tank lip and the body, the metal eventually rusts through, and the tank leaks. From your description, it sounds like this is what has happened.
Once the tank is out, use a hand-powered wire brush to remove the rust. You may see one or more pinholes. If you can't find the leak, put a little gas back into the tank, and rotate it on its side so that the seam is at the low point - I'll bet you'll see gas dribbling out of a pinhole.
You can try having someone weld it, though if the metal is rusted you will probably be throwing good money after bad. You can try that epoxy gas tank repair stuff sold at AutoZone (see picture below) - I have used it in the past, and it seems to work. Or, you could just buy a new tank.