Servo Current Draw When Moving
Assumption: Servo motor in question is a hobby-type servo, not an industrial high torque or heavy duty servo.

The bulk of the current through a (hobby-type) servo motor is the current through the coils of the DC motor typically used within the servo housing. Some current is required by the servo controller circuitry, but that is negligible in comparison.


Current through the servo will thus be high when it is changing position, as the DC motor must turn till target angle is achieved. 
Current is then low after it reaches the new position and stops, as long as there is no opposing torque that the servo must fight to retain the new position.
Current will be high again if a torque is applied to the servo arm... this last can actually be higher than current while moving, up to the stall current of the internal DC motor. This is because the servo controller drives the motor hard to force it to hold the target position.


Thus, depending on current as a means to detect servo traversal is unreliable at best. Instead, a mechanical solution such as a homemade rotary limit switch made using a conventional limit switch levered by one of the servo arms, would be a recommended approach.