here is what may be happening:
Oil in the radiator is usually caused by a blown head gasket between the engine head and engine block. In this condition, the coolant ports and the oil ports are no longer separated allowing to to mix with engine coolant and coolant to mix.
You'd certainly want to drain both the oil and the coolant, replace the gaskets and check the heads.
Does sound like a head gasket. you will need to change both oil & water. Be sure to check timing belts & water pump while your in there. I had the same issue with my 82 mid season last year I was only down maybe 2 days. Head gaskets on that model are fairly simple to change, as long as you are comfortable with a wrench
Is this the same bike that was getting hot 2+ years ago? Did you ever correct that issue? If not, it can cause all of your symptoms. A cracked head, bad head gasket or a cracked block can cause it all.
Hit the breaks on tearing your bike apart this calls for some diagnostics first
You need more info before proceeding there is no water in the oil so there is no harm in a compression test check all cylinders to find the culprit. If all the cylinders test ok with the plugs out with a full radiator pressure test your cooling system look for obvious leaks watch for the pressure gauge to hold pressure if it does not and you cannot locate the leak, crank the motor over and the cylinder or cylinders that are leaking will spray water out of the spark plug hole. If you infact have a blown head gasket send the heads out to a machine shop so they can check for cracks and exsessive warpage.once all this is done if your heads check out have the shop rebuild them. This is not a cheap repair reguardless of whom does the repair so if only one cylinder leaks you may consider only repairing that head. If your coolant system pressure test passes and holds pressure without leaking down repeat this test procedure with the engine warmed up. Note keep the pressure tester on the radiator while running however do not pressurerise the system let this gauge tell you if the is combustion pressure in the cooling system as it will more than likely max out the gauge by itself. All the tools you need can be rented at any napa or autozone type store. I agree with the others that while your in there is time to replace other needed parts.
The next thing to think about is what caused the gasket to blow did the bike overheat due to a stuck close thermostat or a external leak? Perhaps you don't let the bike warm up enough before getting into the throttle. My other thought is if you test everything cold and hot and still do not find a problem buy some oil in water test strips and make sure what you found is infact oil. If it is flush the cooling system completely and refill with just water and run the bike around the block a few times let it cool and recheck for oil in the water if none is found drain the cooling system add the proper mix of coolant and distilled water. At this point cross your fingers keep your rides local for a while to see if the problem persist.