In addition to Rik and Steve's comments, I saw this bit earlier and wasn't 100% sure what you meant but you seem to be saying that the motor is moving within the chassis? In which case, fixing it in position has to be the first thing I'd try. FWIW I had a similar problem with my Wickham Trolley because there is only about 1mm clearance between motor and worm gear and I had to put a second pair of screws through the chassis into the motor mount to lock it.
I also wonder if the heat warping of the body hasn't also affected the chassis, perhaps by twisting it slightly as you tighten the screws? Does it still misbehave with the body off?