Most probably the 9V battery voltage breaks down to 6V when you connect the motors. Such a battery is not really usable with motors and other current-hungry devices. You can try 3 AA cells to power the Arduino directly (5V pin), and also the motor driver. But then add a big capacitor to stabilize the operating voltage of the Arduino board - again the motor currents will result in ripple on the power line.
The motor current is the critical parameter, not the voltage. If you use PWM, the voltage is switched at 480Hz or 960Hz, what should be taken into account when measuring the current or voltage. Does the motor driver board have a current sense output? Then you can add a RC low pass filter to that output, and monitor the average current on the cap.