Big question would be, is it in fact the alarm module or just a dead battery?

I'm facing a similar problem. About a year ago, my alarm kept chirping randomly and I had to pull the fuse. Then one day after putting the fuse back in and the alarm started chirping again the chirp got more and more silent and then stopped altogether.
Haven't had a chance to take a look at it yet but I assume that sounds like the batteries went dead. Why not start with that yourself instead of buying a whole new alarm module?