Ant, Great advice,
The only thing I would not do is hold the bird upside down before it is dead. The reason it gets drowsy is it is suffocating. A chicken has no diaphragm all of the internal organs are pushing on the lungs and slowly suffocating it.
I insist my CX's be held tight and almost hugged until the last possible moment. I want the bird to not go through the discomfort of suffocating and that can release adrenaline into the system which makes the meat tougher and can had an off taste.
That is just my opinion though and each to his own, there really is not right or wrong.
BTW I did not use Ice water just cold water. (not heated)