"First, the alarm should sound
"First, the alarm should sound with the key turned on, as the oil pressure switch at least will be closed (low oil press). Once the engine starts, it should shut off.
Does it behave like this? Or does the alarm stay on after starting?
If the alarm stays on and its MPI it could be any number of things. You will need to get a dealer to read any stored codes and go from there.
If carbed the alarm can be only:
-low oil press
-high water temp
-low drive oil level
-low cooling water pressure (only later units)
-or a short in the wiring
These switches are all NO, so if any one of them closes, the circuit is completed and the alarm sounds. A wiring short will have the same effect.
To troubleshoot I would turn the key on and disconnect sensors one at a time, with the engine not running. If you find one that shuts off the alarm, bingo, it's likely a bad sensor.
Start with the oil press switch first, as it will be closed anyway, and you need to get that out of the circuit to troubleshoot the rest. Leave it disconnected.
Then disconnect the low cooling water switch if equipped; leave it disconnected. Then disconnect the high temp, then finally the drive oil. They are all connected in parallel with tan/blue wire BTW.
If you get to this point and it is still sounding, you have a short in the wiring. That will take some checking with a VOM to trace.
Rod"