Yes, it seems to be a common problem on this engine. For diagnostic purpose I usually bypass the oil cooler from the cooling hoses and you can see oil dripping out from the cooler. Since the oil pressure is higher than the cooling system pressure, oil is pushed into the radiator. Get the oil cooler replaced and the cooling system flushed - it may need to be performed several times and a non-foaming detergent could be a big help to get rid of the oil from the cooling system.
there is no water in the oil .. oil in radiator and hoses. it is not transmission fulid as that would be pink. fix that deal on another car.this is oil .there no water and oil mix in oil pan or dip stick.there is a oil cooler on this car.2004 lincoln ls v8.
Usually means that the cylinder head gaskets are leaking, or the intake manifold's leaking for starters. If the oil cooler you're referring to is in the radiator, then that's for transmission fluid, which could be leaking into the coolant. If you want to find a specialist, here's a directory link for you: https://repairpal.com/directory?address=77630&car_brand_names=Lincoln