1992 Cadillac Deville Cooling system

Hi my 1992 Cadillac Deville radiator had broke. When I say broke I mean the transmission oil cooler broke. So all of my coolant got sucked into my tranny and my radiator fluid is milk brown. So I drained the transmission fluid the same day it happened. So the repair shops say I need a new transmission. Another shop said put a new filter gasket and fluid in it run it 100 miles change the transmission fluid an just keep an eye on it. So I drove it today it doesn't slip or anything it still feels like it was before. But my real question comes in here I put Prestone super radiator flush in the car. It is still milk brown I took a garden hose for 20 minutes and ran it trough the system to clean it out. It was coming out clear now it is back to milky brown with that flush and it is thick. The radiator is brand new. What can I buy or do to get this stuff out of the water system. Someone told me add half a bottle of dawn soAp to the radiator turn the heat on and run it for 10 minutes and then drain it and repay again or until it comes clean. Any suggestion or products on the market that will take care of this for me. Thanks.