There is no thermostat in the cooler. As already mentioned there is a vernatherm which controls oil temperature.
When you drain the oil there may be some oil still left in the cooler, just like there is always oil left on the filter. I don't think this causes any issues as that oil is being pumped through the system when the engine is running.
However oil coolers do need flushing (cleaning) from to time. Perhaps it is clogged.
In general terms there are three possible causes for higher indicated oil temps. Clogged or partially clogged cooler, faulty vernatherm or faulty sensor.
If it's well below the red line I wouldn't panic too much at this stage.
From reading about your high CHT's one other area to look at is your baffling. If the baffling isn't snug and tight at the edges air will be leaking past the baffles. This is air that would have gone through the cooler or past the cooling fins on the heads. This could cause higher oil temp and more likely higher CHT's.
Provided the cowling has been designed properly and has good baffling tight cowling shouldn't be a problem. However poor fitting baffles and or badly worn baffle rubbers can be a problem.
horizon flyer: Lycoming 360 is always one rev away from failure, crap engine.
What a lot of BS. The Lycoming 360 is a very good engine. Perhaps you're getting confused with the Continental IO360?