LeeD is right ... as usual.
Trying to snap your wrist will lead to wrist and arm problems.
It sounds like you are already doing the right thing.
Pat Dougherty, the Bollettieri Academy "Serve Doctor" shows the whole arm is turning, but the wrist is kept comfortably in the same position throughout the stroke.
(You don't need the Leverage Belt to do this, so don't buy it.)