Might want to get a list of the active codes in the ECU. If you don't have a scan tool you'll have to look up the procedure, but it involves pressing a button under the dash and counting the blinking lights on the dash.
I had the same problem with a buzzer. Codes were good, engine temp was good.. I pulled the buzzer
Not saying that's what you should do, though!
In the end my issue could have actually been related to the engine temperature sensor. An aftermarket analog style one was added from the previous owner because the built-in dash temp sensor didn't work.