I've never seen one stay that low, that's pretty bad. I have a suspicion that's not even an oil cooler issue. When I rebuilt mine I did everything like you're planning on doing, new radiator, new water pump (I went with the alliant version with the metal impeller blades), new motorcraft thermostat, new motorcraft oil cooler rebuild etc. Temps stay right in the 185-190 range, both oil and coolant.
I've seen a split that high one time, on an auction truck. I think it came in with a ruptured egr cooler. Cooler was replaced, truck fired up and drove it around. Oil got up to like 275* and coolant stayed down around the 200* mark. Ruptured a new egr cooler almost instantly. Replaced oil cooler and a bunch of other coolant system parts and it was fine after that. Wasn't my truck, and I didn't do the work, but I was there to see the huge spread, it was scary.