Here is what I just tried:
I liked Bondonutz's idea of bracing the motor that he posted above, but I was concerned about bracing it to the chassis. If the chassis bends, what type of pressure is that going to put on the motor and gears? So I wanted to keep it simpler and eliminate the (possible) variable of chassis flex.
The brace I installed definitely cuts down on the amount of flex of the blue motor plate significantly when I push on the motor. Now, the most of the motor movement (which is reduced from before installing the brace) is due to the gray part of the motor mount (attached directly to the motor) separating slightly from the blue motor plate. So, to eliminate this, I would have to do something like Bondonutz did - or figure something else out.