Thank you for the additional information and pictures. It looks like you are powering your shield from your Arduino; please note that the trace that connects to the small VIN pin on the shield is not intended to handle as much current as the driver. The correct way to use the ARDVIN=VOUT jumper on the VNH5019 shield is to power the shield through the larger VIN and GND pins, then the Arduino will be powered through the small VIN pin. Since the Arduino will not draw as much current as your motors, this does not require nearly as much current to pass through the smaller VIN trace.
Could you try removing your Bluetooth module and running our demo.ino example program from our Dual VNH5019 Motor Driver Shield Arduino library? Also, can you post some pictures of both sides of your VNH5019 shield so we can see your soldering?
By the way, I noticed that your power wires going into your terminal block-to-barrel jack adapter are pretty frayed. You should take some precautions against those two wires shorting together (e.g. tinning the wire with some solder, tightly twisting each wire so there are no stray wires, or trimming the stripped portion so all of the bare wire fits inside of the terminal blocks).