A friend handed me an AE-1P three days ago with the opposite problem, shutter won't fire, so I've researched this in the last couple of days.
The AE-1 has a shutter release that has a permanent magnet holding the release mechanism. To fire the camera, current is put through electromagnetic coil around the permanent magnet when the shutter release button is pressed. This cancels the field of the permanent magnet and the shutter mechanism is released. The common problems are that the permanent magnet fails, or the capacitor in this circuit fails, or there is a short or dirty contact in the circuit. Sounds like your permanent magnet is too weak to hold the release mechanism or there is a short that charges the electromagnetic coil as soon as the shutter is cocked. Simply removing the bottom plate, which is easy to do, will let you watch the mechanism.
I've seen people advertizing this repair for something on the order of $80. No parts are available from Canon (from what I've read), so you'll need to rob the magnet from another body, or perhaps replace the capacitor, or clean the switch contacts in the circuit.
I found a free AE-1 repair manual online a few days ago. Can't recall where, and it's too big to email or post. You could Google for it. It has instructions for diagnostic tests and for repair procedures, including disassembly/reassembly. I'll append a URL if I find the source later.
There are several AE-1P publications here, including a parts list, general repair manual, and a troubleshooting guide.