97 Cougar....Radiator not leaking....water pump not leaking....radiator full...getting low coolant warning light and can smell overheating hoses...put on new radiator cap....no change....temp guage shows near normal temps, although higher than usual.....Could it be thermostat ? and where is the thermostat located on 97 xr7 6cly

Open up car bonnet and check your radiator coolant, it’s probably coolant leaks and causing low coolant level.

check for a coolant leak and make sure that the coolant is at the radiator cap or you will get a false low temp reading on the gauge

The overheating probably came from the coolant being too low. The coolant in the reservoir won't go back into the radiator if the coolant in the radiator is too low. Until you get it fixed, check and fill up the radiator every time you want to use it. Overheating is very bad for engines and can cause expensive engine repairs.
Where is it leaking, from the engine or the radiator? You didn't say, but wherever it is leaking is the place to start. If at the radiator, and it is not a loose or leaking hose, I would be looking for a new radiator. If it is leaking at the front of the engine and dripping down from there, you could have a bad water pump. This could be an expensive repair at a shop-not the pump itself-they are not too expensive-but the labor cost in replacing a water pump will add up.

The thermostat would not cause it to leak. But when all is said and done, you will want a new thermostat, after the leak is fixed. They are very cheap and it is just good insurance.

OVERHEATING CAN BE CAUSE BY MANY THINGS LIKE FAULTY THERMOSTAT AND FAULTY RADIATOR PRESSURE CAP,TOO MUCH WATER IN COOLANT SYSTEM NEED 50 / 50 WATER AND ANTIFREEZE.LOOK FOR WATER PUMP WEEP HOLE LEAK, IF COOLANT LEAKING OUT WEEP HOLE WATER PUMP BAD NEED REPLACING.CHECK FOR LOW COOLANT LEVEL, YOUR RADIATOR OVERFLOW JUG SHOULD HAVE COOLANT AT THE FULL COLD MARK IF NOT KEEP ADDING COOLANT INTO COOLANT OVERFLOW JUG UNTIL COOLANT STOP DROPPING AND FULL COLD MARK, IF OVERFLOW COOLANT JUG WAS EMPTY RADIATOR IS LOW ON COOLANT.WHICH CAUSE AIR IN COOLANT SYSTEM KEEPING THE ENGINE COOLANT TEMPERATURE SENSOR FROM BEING SUBMERGE IN HOT COOLANT WHICH TURN ON THE COOLANT FANS.WHEN ENGINE COOLANT TEMPERATURE GET CERTAIN SET TEMPERATURE LIKE 190 DEGREES THE PCM AND COOLANT SENSOR WILL TURN ON COOLANT FANS, IF VECHICLE OVERHEATING COOLANT FANS NOT WORKING EITHER YOU HAVE FAULTY ENGINE COOLANT TEMPERATURE SENSOR CHECK COOLANT FANS FUSE AND RELAYS. YOU CAN CODE SCAN CAR TO SEE IF YOU HAVE FAULTY PCM, FAULTY ENGINE COOLANT TEMPERATURE SENSOR OR FAULTY COOLANT FAN RELAY.IF COOLANT FAN AND ENGINE COOLANT TEMPERATURE SENSOR OKAY,YOU HAVE CORRECT COOLANT LEVEL IN COOLANT SYSTEM, THERMOSTAT AND RADIATOR PRESSURE CAP BEEN REPLACED, YOU COULD HAVE BLOWN HEAD GASKET CHECK ENGINE OIL LOOKS LIKE MILK SHAKE HEAD GASKET BLOWED ALSO WHEN HEAD GASKET BLOWED YOU CAN DRIVE CAR SHORT DISTANCE START OVERHEATING AND YOU LOOSE ENGINE POWER SUSPECT BLOWED HEAD GASKET.

REPLACE THE THERMOSTAT AND RADIATOR PRESSURE CAP.IF COOLANT LOW, ADD MORE COOLANT THROUGH COOLANT OVERFLOW UNTIL COOLANT LEVEL IN THE COOLANT OVERFLOW STAY AT FULL COLD MARK.REASON POUR COOLANT THROUGH COOLANT OVERFLOW MOST GM CARS DONT HAVE RADIATOR CAP.WHEN YOU GET COOLANT LEVEL OKAY, CRANK CAR LET IT IDLE UNTIL ENGINE GET HOT BUILD UP PRESSURE THEN TURN OFF CAR.CHECK FOR COOLANTS LEAKS AT THE RADIATOR HOSES,CHECK FOR LEAKS AT THE RADIATOR, CHECK FOR LEAKS AT WATER PUMP WEEP HOLE IF YOU SEE COOLANT LEAKING FROM WEEP HOLE REPLACE WATER PUMP. IF CAR OVERHEATING WHILE IDLING YOUR COOLING FANS NOT WORKING CHECK COOLING FANS FUSE AND RELAYS.IF FUSE AND RELAYS OKAY THE ENGINECOOLANT TEMPERATURE SENSOR BAD.

IF LOW COOLANT LIGHT CAME ON ENGINE SHUT OFF BECAUSE IT GOT TOO HOT AND OVER HEATED.WHEN YOU HAVE LOW COOLANT.YOU COULD HAVE LEAKING RADIATOR.YOU COULD HAVE LEAKING RADIATOR HOSES AND WATER PUMP. ADD MORE COOLANT IN CAR RADIATOR UNTIL YOU AT FULL COLD MARK ON THE OVER FLOW COOLANT TANK START THE CAR LET IDLE UNTIL ENGINE GET OPERATING TEMPERATURE.RAISE YOUR HOOD LOOK FOR LEAKS THE RADIATOR PRESSURE WILL PUSH COOLANT OUT WHERE LEAKS COULD BE.LOOK AROUND RADIATOR AND COOLANT HOSES.LOOK AT WATER PUMP.IF YOU SEE COOLANT LEAKING AT WATER PUMP REPLACE IT.

Two things would cause this

There is a slow leak and the radiator is low on fluid

Let the vehicle cool down completely and then check the level in the radiator. I bet its pretty low. Top off the radiator with a 50/50 mix of coolant and water. start the vehicle and let it warm up with out the radiator cap on. Keep adding fluid as necessary. While waiting for it to warm up check around for any abvious coolant leaks. Things like hoses, radiator seams, water pump, water inlet/outlet gaskets. If no leaks are found it may be necessary to pressure test the cooling system and even then it might be a slow leak that only occurs when the vehicle is cold.

If the radiator is full than I would suspect that water pump is no longer working or creating enough flow. If the thermostat where stuck you would still get heat. If the water pump is not circulating coolant than you would get no heat and the car would overheat.

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the problem could be as simple as a faulty radiator cap from your description the water that is expelled when it gets hot is not being drawn back as the radiator cools down
could be a leak in the reservoir hose
I am thinking that the low coolant sensor in the top of the radiator tank is getting an air space and that is what is activating the light
I don't think it was head gasket problem as you would be overheating or loosing coolant within days not weeks I would go to an accredited radiator specialist shop and have a pressure test done for leaks in the system including the water pump leaking when running
I am saying that the vacuum created as the coolant cools down is not as good as it should be to draw water back from the reservoir to keep the radiator full to the cap

If it has a coolant level sensor in the radiator yes it will, due to low on coolant due to water leaking from water pump, coolant will leak from water pump if bad, unless it just has bad water pump bearing's, then some time's it will not leak and just make a noise.

1 Inspect Cooling System Mix Coolant level low or flow is restricted. 2 Inspect Belt Incorrectly routed, adjusted, tensioned, missing, or worn water pump belt(s). 3 Inspect Oil Pan Gasket - Performance Ruptured, cracked or leaking radiator hose. 4 Inspect Radiator Cap Worn or damaged radiator cap 5 Inspect Thermostat Thermostat stuck closed 6 Inspect Fan Blade Broken, missing, or defective fan blade(s). 8 Inspect Water Pump Damaged, worn or leaking water pump. 9 Inspect Intake Manifold Plenum - Perform Leaking water pump gasket. 10 Inspect Cooling Fan Control Faulty cooling fan control or circuit. 11 Inspect Cooling Fan Switch - Radiator Faulty radiator cooling fan switch or circuit. 12 Inspect Engine Temperature Sensor Faulty engine temperature sensor or circuit. 13 Inspect Temperature Switch Damaged or faulty temperature switch or temperature switch circuit. 14 Inspect Fan Clutch Worn, loose or faulty fan clutch. 15 Inspect Ported Vacuum Switch Damaged, leaking, or faulty ported vacuum switch. 16 Inspect Radiator Obstructed radiator core or radiator cooling fins. 17 Inspect Head Gasket - Performance Head gasket leaking coolant into cylinders