Pro Mini 5v with Adafruit motor shield v1 power sharing problems
I would check each motor without the motor shield to see if the motors are ok. Check the motor shield motor supply, should be around 12 volts with 8 AA cells. You might run a simple test program to see if the motors run full power forward and full power backward with the motor shield. Check the servo with a simple program like the Arduino Knob program. I suspect the Pro Mini 5 volt regulator is overloaded, causing the buzzing sound, but you have to find out why. One possible reason is that you are dropping from 12 volts to 5 volts via the pro mini linear regulator, creating all kinds of heat. The additional load of even a small servo might push the power dissipation spec over the edge.