Overheated now not running properly

More than likely you have a failing head gasket from the overheating. Your first description of running hot then cooling off as you get moving sounds like the cooling fan isn't turning on properly. Running it that way could have stressed the gasket and the latest overheat finished it off. Do you smell coolant or see any steam from the exhaust pipe? A quick way to tell if there is a problem would be to remove the spark plugs and lay them next to each other in order. If you see one that is much cleaner or a different color from the rest you have a leak in that cylinder. Another way is to use a small camera to look inside the cylinders after you turn it off and see if there is any evidence of coolant in the cylinder, again the top of the piston would be very clean from the steam generated. I would look into this before worrying about the fan. If the head gasket is failing I would price out those repairs first, then decide on repair or replace.