Have you taken your Galant back to who ever did the Timing Belt to test drive the car and feel the 'shift' ? It may not be the trans, it could be a misfire or many other things.
I agree with your mechanic. I used to rebuild transmissions and if there is a heck of a lot of mileage on a trans that has not been serviced for a long time, it IS risky to open it up. This is because the fluid gets thick and dirty and wears the parts. When the new, thinner fluid gets put in the transmission, the worn parts do not operate correctly with the new thinner fluid and the trans, often, will not work correctly.