first off, do not drive it anymore. Get your cooler/radiator replaced. Sounds like the Transmission heat exchanger in the radiator has ruptured causing cross-contamination. Both systems will require multiple flushing. I'm not sure if the cooling system has a bleed off anywhere (I'm just becoming familiar with these vehicles), but flush the cooling system with water only... that will save some money since you don't want to be throwing anti-freeze away. Fill the system, let it warm up to operating temp, shut off, cool down, drain, fill, repeat. When you stop getting any color in the water from draining, you'll be good to go there. The thermostat may need to be pulled/inspected/replaced.
The transmission flush, unfortunately, can only be done with transmission fluid, and may still require disassembly since it's going to take quite a few to get all of the fluid exchanged from the torque converter. I would do this simultaneously as the coolant flush, since you don't want to run the engine with a transmission full of water, regardless if it is only idling or not.
Good luck.