Possibly the new thermostat is bad...........don't laugh it can happen. I'm not sure if you could have installed it upside down. Sometimes that is possible but I am not familiar with your specific engine application.
ON THE TOP OF THE ENGINE WHERE THE UPPER RADIATOR HOUSE CONNECTS...
In a 2002 Mercury Cougar, the thermostat is located on the engine, specifically at the engine's front, near the water pump. To access it, you typically need to remove the upper radiator hose, as the thermostat is housed in a thermostat housing that connects to the engine block. Always consult the vehicle's service manual for detailed instructions and safety precautions when working on the cooling system.
Should be in a removable housing at the engine end of the upper radiator hose
Follow the upper radiator hose to engine--should be located in housing that hose connects too.
Plugged heater core maybe. Maybe the thermostat. Coolant keeps your engine cool under duress. If the thermostat is stuck open, the engine is staying too cool. Your heat is run off of your engine temperature.
I believe that is 190 degrees Fahrenheit ( 88 degrees celsius )
Check your thermostat. Looks like it is stuck open and needs replacing. That could be why your engine is running cold. Atleast its not stuck in the closed position.
how is the best way to remove water pump from 1997 mercury cougar 4.6 engine
On a 2001 Mercury Cougar : The radiator is located in the engine compartment , at the front of the vehicle
Follow the upper radiator hose to the front of the engine. the radiator hose will end up connecting to a metal hose that is attached to the engine. there it is under it.
On a 1997 Mercury Cougar : The engine oil dipstick is on the drivers side of the engine ( for the 3.8 liter V6 and the 4.6 liter V8 )
drain approx. 1 gallon of coolant from the radiator Follow upper radiator hose to housing on engine Remove housing Make note how thermostat is installed Clean surfaces well Replace thermostat and gasket Re-fill coolant slowly to prevent an "airlock" Start engine and look for leaks Run engine to normal operating temperature with heater on Re-check coolant level