Just thinking outside the box…rather than a thermostat issue, could this be a problem with the fan coming on too early?
When you took the old stat out, was it actually in the stuck open position, proving that it was faulty?
You say that with the new stat installed everything seemed normal until the fan cut in and then wouldn’t go off until you switched off the engine,
So just a thought. Dont know whether it could be a sensor issue or something to do with ecu.