Radiator fluid leaks out immediately!

Obviously I can't see it but by your description, that is typical of a leaking water pump. If you have the 3.0L engine, the pump is driven by the timing belt. I don't know that I would trust driving it at all. If you have the four cylinder engine, the pump is external and while it could wobble apart, if you take along a few gallons of water, you should be able to drive it a mile or two, then let it cool down and fill it to drive some more. You can nurse a car a long way that way.

Also, given the age, you might suspect a corroded core plug in the engine block.

That would be more typical of a loose radiator hose clamp, ruptured hose, or leaking heater hose, but those are generalizations. It is common for air flow through the radiator while driving to keep the engine so cool that little pressure builds up in the cooling system, then when you park the car with a hot engine, that heat migrates into the coolant, heats it up more, and the pressure builds for a few minutes. There's a pressure relief valve in the radiator cap to keep that pressure from going too high, but the point is that pressure can be higher after you park the car than while you're driving. THAT can lead to something springing a leak that was about to anyway. The hot engine didn't cause the leak; it just helped it to finally happen now.

Again, those are just guesses. Corroded core plugs in the engine block can seep coolant slowly for a long time or pressure can force them to suddenly pop out. Seepage is much more common. Most of the time the water pump will make noise when it's going out. Worn bearings let the shaft wobble away from the seal resulting in a leak. The seal can also wear out on its own so there won't be any unusual noise.

Heater cores are another suspect area. The brass or aluminum tanks can corrode through but often those holes are plugged by sediment which stops the leak; that is until the hole is big enough for the sediment to blow out. That can also occur when a mechanic performs a maintenance cooling system flush. Flushing the heater core can wash out that debris exposing the hole. Of course the mechanic gets blamed for causing the leak but in reality he just uncovered what was already there. You can identify a leaking heater core by the coolant running out of the air conditioning condensate drain tube. That is under the hood, on the passenger side of the firewall. There will be a four-inch long rubber hose with a 90 degree bend hanging down from the spout. The coolant will puddle just to the left of and behind the right front tire.

I have an '80 Volare that never got hot from normal driving until one day I used it to pull a trailer load of firewood. Without warning it developed a major leak. Found the lower radiator hose pushed off and just a skeleton of rust where the hose clamp used to be.

So you can see there are a lot of variables to consider. Ultimately you're just going to have to poke around and disassemble things until you find the source of the leak. Then we can discuss how to repair it.