The alarm goes off for 3 reasons, overheat, low oil pressure, or engine overspeed.
All you would normally do is disconnect the temp sensor first, then the oil pressure switch 2nd, and see which one silences the alarm. After that, then you have to see if the sensor is good, or if it is doing it's job and reporting that you really do have a problem.
If it's the temp sensor that silences the alarm, then check and see if you are overheating. The engine should be about 160-170F when measured at the top of the cylinder head. Basically you should be able to touch it, and it should feel like a cup of hot coffee. But no so hot that it instantly scalds you. If you are overheating, and you have already done the pump and tstat, then it's going to a bad head gasket. On these the head gaskets and exhaust cover gaskets are 1 in the same. And what would be going on is exhaust gasses are getting into the cooling water side of things, displacing water.
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If it's the oil pressure switch that silences the alarm. Then pull the oil pressure switch and plumb in a mechanical gauge and start and run the engine, and see what your oil pressure actually is. It should be 50+ psi, but if the oil pressure switch doesn't see about 10psi, then it closes and sounds the alarm. If oil pressure is good, replace the switch. If oil pressure is bad, change the oil and filter first and retest. if it's still no good, then your looking at a teardown.