Well, I've drained it once and got some more brown water for my efforts. Then I remembered that when I was running the car and running new water through it to try to get it all out, I didn't have the heater going. I drained it again tonight with the heater running and filled it with water and radiator flush cleaner. It says to run it for 3 to 6 hours and then flush it again. For me, that's several days just to hit 3 hours.
When I talked with my mechanic, he said the same things you guys said about it maybe being the radiator. My car has the oil cooler and the tranny cooler. I see a cooler mounted in front of the radiator. Is that the tranny cooler? Is the oil cooler inside the radiator? If it is, I could bypass it and see if the problem stops. The oil cooler isn't critical. If that takes care of it, I can either just leave it bypassed or put a separate cooler next to the tranny cooler and cap off the entry and exit holes in the radiator oil cooler. If it's the tranny cooler line in the radiator, I guess I'm hosed and will have to buy a new radiator.
I'll get back to you in a few days. Thanks for all the help.