If you are willing to keep trying, I'll keep making suggestions.
You might try removing the 0.1 microfarad capacitors I see on the motor. Why are these there? They present a very low impedance in the rising edge of the PWM drive, and the current spike might be kicking you out. PWM drives like high impedance, inductive loads so that the current is a ramp rather than a spike when a step change in voltage is applied.
When you do the zero ohm test, jumper directly between the LSS pin and the ground pin on the chip. A trace can have inductance which gives a high impedance so that a voltage can appear at the LSS line with respect to the ground pin.
By the way, what is a lipo?

 

A LiPo is a battery...Lithium based...high current.

 

I switched over to this motordriver, and then it works: http://www.pololu.com/product/713

From the A4950; I tried disconnecting the capacitors that are soldered directly on the motor, but that didn't make any difference either.

For this project I'll stick with the TB6612 driver, and maybe I'll revisit the A4950 in a later project. The A4950 was my plan B, after not getting the MC33926 motordriver to work. I'll have to dig more into that driver as well.

Thanks for all your help!