How does the Governor operation work? Flat ground, blades engaged? Not engaged? On a hill? Could be a fuel starvation issue with the carburetor, fuel system. I worked on an old Ford LGT Cast Iron Kohler single cylinder engine that did a similar stunt. Once I adjusted the governor properly, it not only had more power, but more RPM as well. Then it was easy to adjust the carburetor to accomodate the proper governor setting. No amount of carb adjustments would help a lagging governor. I know it's a long shot especially since no one has bothered the governor linkages, shaft, and adjustments. Could also be a leaking carburetor to engine gasket letting it suck air. Puzzling in that it only acts up on a hill under load. How does it do on a hill without blades engaged? A governor lagging on response will cause an engine to lug and work itself to death by holding back RPM when it needs it most! Concerning the head replacement, I've never heard of a good head causing oil consumption like that. A bad head, warped, or blown gasket would cause loss of compression and power. Might not even run.
If bad rings are pumping oil with a lot of blow-by, it should have a cloud of smoke coming out the muffler. You said it was fouling plugs too and that would in itself cause an engine not to run at top level due to oil fouling. Something is causing that oil fouling besides the head, possibly a broken ring, oil ring bad, could be worn valve guides making it pump oil as well. Is it getting good compression?
