I will give this a shot,if that thing is running as long as you say it is it has to be getting some form of cooling water or it would be seized up by now I would think.
make sure the tell tale hose isnt plugged with leaf cutter beetle eggs and tiny chunks of leaves left for food for the larva,or dirt and sand from running in shallow water.
I will admit I am not real familiar with the injected Evinrudes but it almost sounds like an injector is sticking open when warmed up and flooding the motor.
I would suggest taking it in and having it scanned when its cold to see what the sensors are reading versus when the motor is warmed up good as sensors can change quite a bit when they are failing with different temps.
if and when you decide to take it in,make sure they take it out and run so the issue can be duplicated as the computer may not hold old codes.
keep us posted on whats found out !!