Maybe a sensor or you still have a cooling issue?

But if I had a cooling issue wouldn’t it be present on the gauge? The temp was at 150-160 when the alarm went off.

No, not necessarily
Don't know the serial number but being a 2000 model I'll guess The temp sender signals the ECM and the ECM sounds the alarm for an over temp. The gauge has another sensor which just provides info to you and does not talk to the ECM

That was my thought. So with that being said if the sensor that talks to the ECM is faulty that could trigger the alarm. Assuming the sensor that talks to the gauge is good, the gauge was reading 150-160 went the alarm went off.

Use a IR temp gun to verify temp at the thermostat housing