My alarm issues have been:

Alarm randomly going off regardless of car on or off, doors locked or not.. as in the alarm would go off while driving

Replaced the siren, alarm has never again gone off while driving

However, the alarm does sometimes go off when the car is off and locked. This I have no solved but suspect it may be from the module in the trunk though I've not been able to find evidence of water in there.. however, when the car is unlocked (and off) there is no issue

So suffice to say, if your angry PMSing cricket sound is gone but lights are still flashing it may be related to the siren itself causing issues with the whole system. This is the "easiest" to replace so let's hope its that.. added bonus: replacing the siren to a newer than 2005 version will give you a little chirp to acknowledge you've locked the doors!