The relays are in the "door lock timer" module. The wires going to the lock/unlock switch (near the gearshift lever on 94-up) and the security module are all low-current control lines (~50mA).
The security module contains the wireless receiver. If your module is missing or defective, a keyfob transmitter will never work. The fact that you're not seeing the red Security light turn on/off when you push the button on the dash suggests that your module is bad or missing.
Even without the module, your passenger door is supposed to be locking and unlocking slaved off the driver's door. Since that's not happening, you have more problems than just the security module -- either the door lock timer, wiring, or (most likely, due to corrosion from water) contacts in either of the lock actuators in the doors.
I recommend first looking to see if you have a security module. You'll need to remove the glove box. There are threads that describe the procedure (lots of screws to remove). Search.
With the module unplugged, connect each of the pins for Lock and Unlock to ground and see if at least one of the doors' actuators works.
If not, then you'll need to open up the doors and dissect the actuators.
Once you get that sorted out, upgrading to an aftermarket alarm is probably the better way to go. The OEM modules are prone to die. Just not worth the cost IMO.
All of the signals you need are available on the security module harness. You can connect your new alarm there (the "brain" can be somewhere else under your dash, although the OEM location is fairly secure since it's so hard to access). See this thread:
After market alarm install document
I have a connector that plugs into the factory harness. E-mail me for more info.