My 2003 LeSabre over heated

My 2003 LeSabre over heated Friday, I had it towed to my mechanic and he replaced
a plastic connector for the cooling system, he replaced it with a metal one. He was
concerned about the color of the coolant=Brown and very mucky. He changed
thermostat and told me to come back in a week to check coolant color. He was
worried that the transmission cooler he said was inside the radiator was leaking. I
didn't get 3 miles away from his shop when the car was overheating so I returned.
When we popped the hood the engine was very hot. The coolant over flow reservoir
was boiling coolant in it and it was brown again.
I left it there for the weekend as they were getting ready to close. I showed up Monday
morning and he did a test by putting a clear cylinder with blue liquid in it where the
radiator cap goes and said we will start the engine and if it stays blue were good if it
changes color were screwed. It changed color. He said it meant that exhaust was
getting into the coolant. It’s probably head gaskets.
His take on it is that the transmission cooler inside the radiator leaked and the higher
pressure transmission fluid caused the overheating and the plastic coupling to brake
and ultimately to cause the head gaskets to give way.
He doesn't recommend I repair it. He recommended that I just sell the care to
someone who does this kind of work. And I have also been told by some others that
replacing the motor and radiator is a better way to go.