To overcome my early reluctance to shoot a deer I decided to use enough gun to kill it very quickly and humanely. I've found a .300 Win Mag and a .270 Win kills them instantly and they don't know what hit them.
The TV shows all show them running off after the shot. I NEVER find this to happen. Mine have always dropped on the spot. And I've shot dozens and dozens of deer. Of course I always hit them in the lungs/heart or sometimes the neck if that's the only shot I have.
Just a thought for you.