Your cousin needs to practice several boxes of shells at the range not using the bench - mimicking actual hunting positions. He needs to practice as if he's actually shooting at a deer - every thing that we do when we draw down on a deer that appears out of the brush. See the deer. Quickly and quietly flip the scope covers if needed. Slowly but surely get a good position and rest on whatever is available. Acquire the target - the RIGHT target!!! (not just getting on brown!). Flip the safety. Exhale, squeeze. He needs to practice this slowly and carefully in a variety of positions and gradually speed up the sequence when he can do it accurately. In otherwords, he should go through the routine and fire as quickly as he can, but only as quickly as he can do it smoothly, quietly, and ACCURATELY.

If need-be he should use life like targets.

Anything that makes this all routine and almost without thought - but at the same time, a good aim and accurate shot.

Sounds like he's just not practiced and gets flustered.

AS for loosing the deer- he also needs to learn the importance of letting a deer be after the shot. At least 1/2 hour for a good clean chest hit. At least an hour if not two for a leg shot like his. A leg will bleed out and fairly quickly if the femoral artery is hit, but they will also hang in there and run for quite a while afterwards if pushed, but if not pushed will often lay down after just a minute of running or trotting.