Tying the board connectors together via their grove connectors sis not help much. In addition, I desoldered the reset switch on the motor shield and replaced it with a .05 uF disc cap. That helped more, I can run the motors in reverse at half speed now.
Next step is to put some caps at the connections to the motor itself. I put a couple of caps at the motor connection terminal block, across each motor line, but that had no effect.
I am even thinking of building a “bypass shield” - a shield with nothing but caps and zeners on the Arduino power and signal lines, along with big conductors for 5V and Ground, to be placed between the computer and the shield stack.