The transmission seals are the likely reason for the oil leakage. They deteriorate with age. Not a real big job to replace, but does require splitting the tractor.
If it's a 2 stage clutch, there will be 2 seals to replace. While in there it would be a good idea to take a look at the clutch, release bearing, pilot bearing, and flywheel starter gear.
If you take the clutch out, and it is a 2 stage, you MUST have an alignment tool to get the clutch discs in line to receive the transmission shafts on reassembly. There are also initial adjustment to the clutch assembly before going back together. A shop manual is a very good investment. Stabbing the engine back is difficult under the best conditions, damaging and dangerous otherwise. Best not attempted without the proper equipment to support the engine and back half of the tractor, and a smooth concrete slab to work over.
Assuming this is a gas engine, the governor may be having some mechanical issues. They have a reputation for the ball cage to fail. This is located under the timing cover. But before going into that, try to go through the inspection and adjustment procedure.
For now, lets assume the governor itself is good, and something is just out of adjustment.
Some governor theory:
Incorporated into the linkage is a "governor spring". It's job is to pull the throttle (on the carb) open. Moving the dash lever to the fast position tensions the gov spring. The further it is pushed, the more spring tension, the more RPM.
Inside the governor housing is a cone shaped disc. Inside the disc is a set of spinning "weights" (steel balls). The faster the weights spin, the more force they exert on the cone, pushing it forward. When the cone pushes forward, it moves the governor link, which pushes the throttle closed.
There is a constant balancing act between the spring pulling the throttle open, and the weights pushing it closed.
Let's get started...
With the engine off, set the dash lever to the idle position. Looking at the carb, locate the idle stop screw. The stop screw should be against the stop block, as in the carb is in the idle position, throttle closed. For now, I am assuming the idle adjustment is good, as your post didn't say it was running away or idling too fast.
Next, with the engine off, move the dash lever to full speed. Look at the carb, the throttle should now be wide open, approximately 90* from idle stop position. If it is not, something is binding or out of adjustment.
Start the engine.
Set the dash lever to idle. The engine should be at idle speed.
Bring the engine speed up. Hopefully you have a way to monitor the speed, either the proof meter, or a hand held tach. The max speed should be 2200 RPM. If the RPM is low, look up at the top of the engine, just above the governor. There is a U bolt clamp. Loosen the nuts, turn the lever (without turning the shaft) until the RPM comes up to 2200, tighten the nuts. Cycle the dash lever a few times, watching the RPM, it should return to idle, repeat to 2200 each time.
With the engine running at full governed RPM, hand turn the throttle on the carb to idle, let the engine wind down, then release the throttle, watch what happens.
The throttle should instantly spring wide open, then return as the RPM comes up, and level off at 2200 RPM.
If it will not, something may be wrong inside the governor. You can try adjusting the length of the governor link (the rod between the governor and the carb). But unless something has been previously adjusted or carb work done, the length didn't change by itself.
Inside the governor, there is a cage that guides the spinning weights. The cage is known for coming apart. When it does, the governor becomes erratic, but worse, pieces of the cage can fall into the cam/crank gear, and fall into the oil pan. There is also a pin that can shear where the arm attaches to the shaft.
Lets hope none of that has happened!
When you look it over, make the adjustments, let us know how it goes.
Governor Adjustment