another point, I think, Transmission fluid is not cooled in the radiator, its in a section of the AC cond. so a bad rad will not do this, Looking at the soup in the pan, I would bet a good amount that it is water in the trans.
As others said, I would find a shop with one of those transmission fluid exchange units, I think they flush everything, including the converter which you can't get at as well as the cooler.
I remember back in the stone age they has a drain plug.
I would block off that value, or remover it, but I have never played with mine.
Sorry , a bit late to the party .