Got an oscilloscope handy? It'd be a good idea to see what happens with the servos control signals when the motor is turned on. If you don't have access to a proper one, a 'arduino scope' might do the trick.
I'd say your problem is noise on the signal lines, motor and motor control circuitry is notoriously noisy and might be breaking the threshold for the rising and falling edges of the pulse. The hobby servo control logic is meant to avoid this, but you never know. Eliminating noise on stripboard might be a bit hard, but good grounding and physical separation might do the trick.
Looking at the control signals might give you a clue to anything else that's wrong with your code, i.e. servo timing changing when motors are throttled up might indicate faulty timing code.
Alec