Did you not change out the poppet? Cheap insurance every time you change out the thermostat.
With the engine running and the alarm going off, reach around and feel the water coming out with your hand - is it barely warm or is it hot?
It is possible you have a continuity issue in the wiring from the sensor to the alarm. Check for ground or broken wires, or bypass with a known good wire for a test. Or, possibly it could be a bad temp sending unit to the alarm.
Safe to say, forget the oil (Synthetic or otherwise) as being the issue.
If it were me, the next move would be the POPPET. Study a diagram for your engine as to the orientation of the poppet to the gasket - some go under the gasket, some go over the gasket, on some the thermostat goes one way and the poppet goes the other (Under/over the gasket, that is). Plan your work! Good luck!