I don't know why you would try and check the oil of a running engine because it is not possible to get an accurate reading.
Inside the crankcase, oil is flying everywhere, thanks to all of the components that are spinning. Crankcase pressure is also common. If you want to know the true health of your engine, then you need to conduct a leak-down test as well as wet and dry compression tests. The results of those tests will give you insight as to how much wear is on the engine parts and which engine parts (if any) are worn to the point of being a problem.
Rebuilding an Onan engine has never been inexpensive and today the cost has risen to well above the $1000.00 figure, depending on how much of the work is done by the tractor owner. That said, a competent engine builder/machinist should be employed to assess the block, do the machining necessary and then re-assemble the basic engine. Adding the heads, flywheel, sheet metal, starter, coil etc are all simple tasks along with putting the completed engine back into the frame.
I'm not saying that you need a rebuild. What I am saying is that you need to find someone to conduct the above tests to determine whether a rebuild is needed NOW or in the near future. Until those tests are made, it's anybodies guess whether the problems you are having are major or minor. A leaky intake manifold can cause all kinds of havoc and that is a common problem. Issues with the governor spring can also cause the "hunting" you describe.
Engine diagnosis is a step-by-step process and it begins with making sure that the valve lash is set correctly, the points and plugs are gapped to spec and that the carb is functioning as it should.
As for the gear changing issue from Hi to Lo range, just rock your body back and forth when sitting on the tractor and it will also rock. it only takes a couple of inches of movement to synchronize the gear teeth so that they mesh. Using the travel lever to do this is not always advisable because you run the risk of chipping the edges of the teeth. If that happens, the chips can get pulled into teeth of the Hi or Lo range gear and smash those gears when they reach the cluster gear.