Saturn :: 1999 - Security System Chirps Intermittently - Windows / Sunroof Leaking?

My 1999 Saturn SL2 has had issues with its security system for years. It "chirps" intermittently, sometimes to the point of killing the battery. I've taken to unplugging the battery when I won't be using it for a few days so that the chirping doesn't get out of control. Often I'm not able to start the car right away, because the security system has been activated somehow. When this happens I have to unlock and lock the doors until the car starts. I used to be able to improve this by using the keyless entry system, but this very recently stopped working, even though the batteries are new. The security came with the car, and was definitely not installed after market.

My car has also suffered from water damage for a long time. The trunk initially was leaking. Since then the sunroof, drivers side window and rear window have all become leaky. I have fixed those with caulking, but nobody has been able to fix the trunk leak. (really quite significant; when I lived in Portland I sometimes had to siphon out several inches of water from the trunk). My theory is that the electric system might have water damage, but I've brought the car to several mechanics, including electrical specialists, and nobody has had any answers.