To keep it in the same position, it does use the "current draw", but that's not including if there is a Load on the servo. (Obviously, it will require more power to keep it in the same position if it has some weight to keep stationary)
Even when a servo is in the same position, you've got to update to the servo ~20ms ish. 
But I'm going to agree, if it's only an issue when the motor comes on, sounds like a noise issue for sure. (Don't rule out anything.. still might be a current draw issue)
Also servo's won't act the same if you're using an external crystal that's not exactly 16MHZ, or grounded with the capacitors right.
Just something to keep in mind. Shouldn't have a problem with that on any Arduino board, but anything's possible!