Re: Transmission Control Module & Shift solenoid issues
IIRC there is no separate transmission control module; it's integrated in the car's main ECU. I would expect an ECU failure to cause problems other than just a couple indicator lights failing to work and the CEL and AT lights to come on. The price ($1100) sounds about right for a replacement ECU though...in the end, my gut feeling is that they didn't diagnose your original problem thoroughly enough and suggested replacing the ECU on a guess.
Also, assuming the car was driving properly otherwise, a shift solenoid wouldn't be the problem. Shift solenoids are what makes the ATX shift through gears.
I'm thinking you have a problem with a sensor somewhere, or a wiring problem (pinched, corroded, etc.). Since the CEL and AT indicators are coming on, the tech should be able to read the error codes that the ECU is throwing and use them to diagnose the problem. Since it's a Mazda dealer you took the car to, hopefully the tech will finally have the intestinal fortitude to call the Mazda tech hotline for help with diagnosing. You might want to politely suggest they "call to see what Mazda thinks is going on with the car" to get the point across that you're tired of them guessing as to what the problem is.
Incidentally, how many miles are on your car, and how often (if ever) have you had the transmission fluid changed?