I had a Carry-Cool way back in the early '80s too. Don't know what caused the water leaks but certainly loose accordian panels could allow rain in.

I will note that I've used several of the smallest modern window A/Cs over the last five years, mostly Frigidaires, and besides likely being more efficient, they're also considerably smaller and lighter than those "portable" GE units from long ago, even though the new ones aren't billed as being portable (they're about 16"w x 12"h for the main box). I don't find them any more difficult to install or remove than the old GE was - there's no handle on top, but their small size and relatively light weight make it easy to manuever. Given that the new ones start at about $99 for the cheapest ones, wouldn't be so bad IMO if you needed to replace it.

When the AC does dehumidification, the water removed from the air drips onto a tray in the bottom. That's also where some portion of the rain ends up.

The tray has holes that let the water fall out. If the holes in the tray are plugged up -or- the entire unit was installed tilted towards the inside of the house instead of slightly sloped towards the outside of the house, you'd get that problem.

Some window units don't have drain holes in the outside condensate tray. The water is collected and blown against the condensor by the fan to help with heat dissipation.

If it's any help. here a photo of the bottom of the Carry-Cool. O don't see any holes into which the moisture might be drained.

Man, that's a dinosaur! Newer ones will be much more energy efficient and give you better cooling for less money. You should replace it for that reason alone.

Here's photo of it from the outside, showing how it's positioned.

I think ones where the water is intended to help with cooling are a bit more modern that this particular one.

It's pretty much got to be the bottom tray. The exterior picture isn't very helpful. The exterior sill has a shape cut into it (plus the angle of the photo) makes it impossible to tell what's going on.

There are supposed to be visible weep holes in the storm window frame at the bottom. They are to let any moisture that forms on the windows out, but would also help turn small leaks into lakes. The storm windows should be the last thing out, so also strange that you appear to have wood trim beyond the storm.

The vinyl panels on the side would never have keep rain out of the area between the wood sash and the storm window frame. I doubt this is rain - it would need to be pretty constant rain.