It's probably rusted through. You are on cape cod, so that is about as bad as it gets for rust; you're boned.
Click the project link in my signature, I was in the same boat. If the back face of the axle housing is getting wet, then you are either leaking out the fill plug (should be obviously centered on the plug), or you are rusted through. A leaky pinon seal will leak onto the front end of the diff, and a leaky housing seal will drip to the bottom. You are probably leaking through the housing, and it's soaking into all the rust, which is why it isn't just dripping actively.
You can take some spray cleaner and blast all the oil off, then check it again to see where exactly the leak is, or you can just take a twisted wire brush on an angle grinder to the housing and find the pinholes yourself.
I am sorry to have to welcome you to the club...