The fan runs for a very short time every few minutes.

I have been a natural gas service tech for 30+ years and have experienced this problem several times. It is caused by the gas control not completely shutting off and causing pressure to build in the burner tube . This causes the pressure switch to start the exhaust blower, makes the second pressure switch, which opens the solenoid valve on the burner tube. That relieves the pressure and the unit shuts off. This happens when the water is at the proper temperature and the thermostat (gas control) is not actually calling for heat.

Solution to the problem is to replace the gas control. This can be verified by connecting a manometer to the solenoid valve on the burner tube where the pressure switch is connected. It may take a few minutes for the pressure to show on the manometer as the leakage through the gas control is very slight.