Hi, I would also start with the heat exchanger/oil cooler first, have had this before. You would need to replace it and do several good coolant flushes.
Actually it can, from experience it would make more sense that one would first start with the oil cooler, end of the day it is your choice. Maybe see what the other members have to suggest..
Is your car Auto or manual? If auto then there would be two oil coolers. One for gearbox and one for engine. Most likely its one of the oil coolers better known as heat exchangers. Gearbox one is at the bottom of the radiator under the expansion bottle, engine one is by the oil filter.
I stuck a hose pipe into the expansion bottle. Opened the tap slightly so the water just trickles out. Started the car and let it idle until i saw no more oil flowing out. I did this method 3 times when the motor was warmed up.