Mazda 3 2006, auto transmission, starts, revs, but won't move

What is your question?

While test driving a 2006 Mazda3, it seemed slightly hesitant to go into each gear, which would cause it to jump slightly. Dealer's mechanic drove it and said it's normal for Mazdas. After buying the car, we realized that there was an "AT" light. The very next day the car would shift into reverse but it wouldn't move backwards. Later that day, the car stopped moving forward. We added some transmission fluid (this was full while on the dealer's lot) and the car would move forward, but not at normal speeds. It was very slow and sluggish. That evening, again the car would not move forward or backwards. I noticed when I put it in reverse the car would try to roll forward slightly. I also noticed that when I put the car in neutral it tried to go forward. Dealer's mechanic replaced the transmission and 1 day later the same began to happen. Also replaced the TCM and 2 speed sensors. Nothing is helping. Mechanic is ordering another transmission. I am not convinced that 2 transmissions are bad

What makes this problem better or worse?

It got worse as the engine warmed up.

How long has your 2006 Mazda 3 had this problem?

1st day, then undriveable for a month.