We have our servo motor controller daisy chained to one of our DC Motor Controllers.
We have two servo motors on our robot and both of them are defined independently on our Schematic editor. We have assigned separate buttons to each servo motor BUT when we press button 6, both servo motors move.
It is a bit of a bother when using the Schematic Editor to resolve servo postions (unplug ones you don't want moving if it is a critical issue).
Are you seeing the same behavior when you are in the middle of your driver controlled (teleop) operation, or just at the very beginning?
Is this a critical issue for your autonomy? If so, then make sure to initialize all the servos (we put the Move Servo call in a while loop that runs exaclty ONE time as soon as the program begins - this is allowed.) [how to do this is addressed in the thread linked to above]
You should find that even without a lot of effort the actual operation of servos during teleop and autonomy will resolve itself after that initial twitch.