Fired it up.  Plugged in a skill saw with a zip tie on the trigger as a load and plugged in my kill-a-watt as an output indicator.  With the saw running, it would run about 56Hz and put out 115 volts.  With no load, I adjusted it to about 61Hz and 134 volts.  I couldn't get the throttle under load to go any higher.  There maybe other adjustments, but it was getting dark and they are all under the hot muffler.

Such generators are notoriously known for not as good voltage and frequency stability, they are really designed to power just such power tools, not anything more sensitive.
What was the load when the saw was running?
The 56 vs 61Hz difference sounds quite a lot (about 10%) for a governor sag, I don't think the difference uses to be that big, even with a lawn mowers (some 600W Honda and few cheepeese gensets with BS and Honda engine clones, one ~70cc Honda lawnmower). But the service manual will tell more...
Are you sure you were adjusting the governor (usually on the crank shaft case) and not the "idle" adjustment screw (on the carburetter). The "idle" adjustment is not that much useful on a gensets (the engines are designed as universal, so are usually equipped with that), because all the time the throttle should be controlled just by the speed governor. So you should make sure it does not interfere with the range the governor may use. You may test it by manually pushing the throttle to rev it up a bit and then release, so let the governor to settle the rpm back. The engine should slow down back to the nominal speed pretty quickly.

115v sounds pretty good, though 134v sounds a little high! But it's not inverted like a 2000w Honda either, so....of course the voltage and frequency will be a little off.

OK Icefog, here's an interesting one to add to the generator voltage/hz issues topic.  For years I've seen generators (mostly Stamford generator heads on Isuzu diesel engines) that always "flickered" lights (Like a rectifying fluorescent but slower).  I always thought this was bad hertz (too slow).  But I knew a case of someone's (it was a Lister though) which had like 62Hz...in fact THAT sucker made LPF fluorescents strobe so annoyingly I scored several vintage 1970s Sears 4' shoplights, albeit with their original ballasts gone and replaced with crappy LPF residential ones.  I'd asked people why does this happen? for years.  Finally a guy at the AvTec campus in Seward told me yesterday...apparently the voltage regulator goes out in those generator heads and it thus constantly "searches" for the right voltage.  And said yeah it makes fluorescents flicker bad.  The room the giant villiage electrical plants were in was lit by 2X40 vaportights which "flickered" as all HPF RS ballasts and 40w lamps do, and I said yeah they don't flicker quite like that, I know fluorescents well and it's slower than that. 
He also said the hertz can be adjusted by adjusting the governor...whatever that is (I'm mechanically ignorant).
But 134v sounds REALLY high!

Governor = the mechanism which controls the engine speed. The engine speed is directly related to the AC frequency.

OK that makes sense.  But the flicker I'm describing was "slow" (I'd say 10 or so times each second) so the voltage regulator thing makes sense.

Or the governor mechanism has either some excessive friction or play, so the speed oscillates, what means not that regular operation. and that you should recognize in the noise the engine makes, or at least you will see the throttle linkage shaking at that 10x per seconds.

The governor is in fact a feedback loop as any other, where when all the parameters are not aligned well, it turns into a kind of an oscillator. And the excessive friction or play in the governor linkages (from the centrifugal sensing part on the crank shaft to the throttle valve on the carburettor) or so are aspects deteriorating the parameters of the system and usually a result of the machine wear.

Don't run this one and haven't in months now (broken exhaust system) but it's speed sounded like any other diesel like that...

If you mean idling diesel cars, except the modern common rail systems, what I've seen the operation was quite irregular (the reason is, the fuel dose required for regular idling by firing all cylinders is below the minimum the injector system was capable, so it operates more in a hit-miss mode).

And if it is diesel engine, the governor uses to be an integral part of the high pressure injection pump assembly, so no linkages or so would be visible.

It sounds constant and is much faster than any diesel truck idling.  Basicaly it's set to a certain RPM and left there.

If the firing is really regular, then indeed the governor would be OK.

Something else came to mu mind: When browsing for some documentation, I've seen just the gasoline versions with 2-pole generator, that means the engine should operate at 3600rpm for the 60Hz output. If the diesel version uses 4-pole generator, it would have to operate at 1800rpm.
But if you think about how the 4 stroke engine work: It fires every other stroke and just before the firing, the piston "consumes" quite significant momentum for the compression. That means the shaft speed becomes "modulated" by half of the rpm, so 30Hz for the 3600rpm engine and at 15Hz for the 1800rpm one.
As the generator is directly on the common shaft, such speed modulation could cause some flicker. And the 15Hz may be pretty well the "about 10x second" you have observed...

Hmm interesting...In that case there's nothing that can really be done about it then I assume?

Indeed, I have no idea, how to supress the 30/15Hz flicker either.
Well, beside using heavy flywheel and heavy, rigid engine mount...

Interesting nonetheless though.  But to me the voltage regulator honestly makes the most sense...except I know of a brand new unit that did this from day one.

If the engine is 3600rpm, it runs regularly and the flicker is not faster than the ~10x per second, then indeed, there is not much else left...
But if the voltage regulator does that, it will be just it's design, I could not imagine any real defect yielding such flicker (except the mechanical ones, but those would manifest by an irregular engine firing or shaking)...

I THINK this one is 1800RPM but I'd have to look at the nameplate...