You may be heading in the right direction. I know you said its “electric type from Cat” but its still likely either actually made by Woodward or Heinzman. I didn’t think Cat made their own electronic gov. But either way, you should be able to put a meter on the speed setting transmitter input to the elec gov to check (depending on type) the mV or mA signal to the governor. See if this changes before/after putting the shaft generator online, and again after the couple hours.
Most generators I’ve worked with will default to a slightly higher RPM either when taken in local manual control, or when the electronic governor has failed in some way. This was intentional so that the generator would be running slightly over 60hz and ready to take load. It was also a good indicator of a problem with the elec gov.
Something else to consider is the fuel rack actuator on the engine. Again dependent on the type, but if it is in anyway hydraulic or reliant on oil, this is a component that should be inspected.
As @MickAK mentioned you should check any mag pickups you have. Usually when there is an issue with a dirty/greasy pickup or damaged tooth it results in fluctuations not slow steady speed change. However if this is the only source of your speed input you should verify it anyway as it is easy to check.