torqueflight trouble!
Unfortunately I've never worked on a cable shift TF at all but I have read that if the cable is out of adjustment it can do crazy things like pulling in drive. I don't have a service manual for an early A so I don't know the cable adjustment procedure but that's the first thing I'd check out. Maybe try a search here on the board or a google search for torqueflite cable shift adjustment procedure once you get the radiator problem solved.
Another way to solve the radiator problem without replacing it is to block off the cooler in the radiator and run the trans. solely off of an auxiliary cooler. You can get them from a lot of auto parts stores, Summit, JEGS, etc... Get a pretty good sized one cause heat is a torqueflites enemy.