Headlight washer system leaks out all the fluid just sitting without using the pump button

The main culprits for overheating in any cooling system are: malfunctioning fan, bad water pump, coolant leaks, stuck thermostat, bad radiator cap seal, head gasket leak, contaminated fluid or bubbles in the cooling system.

You can eliminate water pump and thermostat!

this indicates a leak back problem if on a fuel injected system or a leak in the carby system
if a manual pump it would indicate that the pump valves are faulty and not sealing properly and on an electronic system, it indicates a faulty pressure regulator , failing fuel pump, faulty filters , leaking hose connections

Check ALL of your fluids - Power Steering, Brake Reservoir and transmission. You could have a cooler line that is leaking.
Does your AC still work?

Is there a possibility that its from another vehicle?
- Something like apartment complex parking?

If no, look for drips off of components in your vehicle to isolate the source. You may need to wipe down the under carriage and start the car to see where the leak comes from. Use a flashlight to help find the drip(s).

Is your wiper fluid level good?Look for leaks at pump,it's connected to container,also check your hoses leaving pump to windshield,could be loose or binding.

Have someone press or hold the washer switch and listen underhood for a small pump sound {Hum}. If you don't hear a small hum you probably need a new washer fluid pump, pretty common for this old of a car. Imagine for the last 15 years sitting in wet fluid,sometimes drying out, getting refilled, turned on and off, on and off that little pump had it's day and retired! Most auto parts stores sell inexpensive replacements. Easily come out and go back in in reverse order. Pumps are usually mounted underside of fluid tank,just follow any visible washer hose.

It sounds like the problem could be either that the flexible hose that connects from the engine bay to the hood has become kinked or squashed or that the pump itself is not providing sufficient output. For the kinked / squashed hose, use a torch to observe what happens to the hose as the hood is closed. You may have to re-route or twist the hose on the pipe so that it sits without any interference from any other items in the engine bay. If the hoses are OK, then the pump could have some restriction at the inlet or is not running at full speed. If the washer fluid appears cloudy, then there is probably some algae blocking the inlet screen on the pump. To remedy, remove washer bottle and flush / wash to remove any slime then replace and refill, preferably with some washer additive. With hoses and restrictions eliminated and flow is still poor it could only be the pump motor running slow. This could be caused by a leaking seal on the pump shaft allowing water into the motor causing corrosion. You could try dismantling the pump and applying some water resistant grease to the shaft seal but it would only be a temporary fix. Best option for this fault would be to replace the pump motor.

check lose hoses or the windshield washer fluid bottle for cracks.

Generally.....3 things happen.
The washer fluid plastic bottle may be cracked....not likely.
The pump (and or gasket at the pump) may be leaking ......30%.
The rubber hose pushed onto the pump to direct the washer fluid up to the window may have popped off........70% ......most likely.

It needs a master cylinder. The only other thing it could be is if you have an exhaust leak, boiling the brake fluid, but this would happen as you drive it, not just sitting there, without it running

the lines could be dry rotting or just a pin hole can do it but it does sound like somethings loose on the reservoir