Yea, those factory alarms can be a PITA, but pretty much only when people don't know how to use them... please no offense but there is no reason to hate the factory system.
I'll start with the worst case scenario: You go to the junk yard and get an engine computer and body computer from a closely matching '91 car. Install them simultaneously and remove the alarm module while the battery is disconnected and it should fire right up and NEVER bother you again. Make sure the junk yard car doesn't have an alarm. Just simple removal of the alarm module at this point will not get the car started.
But I would focus on getting the existing system deactivated. You tried deactivating it the right way by turning the lock cylinders in the doors (make sure to try both ways a couple times in both doors) but it is common for the little switches in there to fail, so the solution is usually to go to the passenger side (the less often used side) and deactivate it there but that switch can fail, also.
If you disconnect the battery, and re-connect it, all the parking lights around the car should flash and so should the alarm indicator light inside the car. It does this to alert the owner that there is a factory alarm system present and it needs to be deactivated before the car will start and stay running. If working the lock cylinders doesn't work, I would remove the door panels and find the wire that goes to the door handle and manually ground it. This is all that turning the key does. That should get you started again.
This is cool because I have NEVER seen a K-based car with the remote entry fob before. What does it look like? Where is the reciever module at? I'd love to get one for my '92 GTC convertible but they are tough to find. People wonder why I lock the doors when the top is down and they sure do jump when they find out why! Its got the factory alarm in it. =D