I'd get a gauge on there to tell what you've got.
Yes, that's the shaft I'm referring to and yes you need to be operating to get balance between spring and fly weights to have control of the shaft rotation. The linkage in these governors is a bit tricky to change lateral motion of the governor into shaft rotation.
There is also a bypass valve and a check valve under the large cover. Have you checked their operation? Also check valves in the priming pump that were previously mentioned.
I'm now out of ideas. Let us know what you find.

One more idea...could suck air in around the priming pump shaft. Can't find a cross section detail of how it's sealed.