Hi Sam,
yes I dry off the veneer after a good soaking, then apply the kwik grip, follow the instructions and then put a flat sheet ontop of the veneer and clamp.
For the body veneer I'd cut the rough outer shape which you have done. Don't cut any internal shapes as the final position of the veneer you won't know.
Once it's dried you should be able to feel where the edges are. Get a sharp blade out and start cutting the cavity, very carefully