With turkey, I make sure my boot is firmly planted on the neck before trying to grab the legs. I usually do a bit of a grind on it if the gobbler is still active. After that, it's just a matter of "lift and separate." I put it back down while I pick up my stuff.
With deer, the first thing I do is check the legs. If the legs are drawn up under the deer, it's still alive and it gets a round to the chest. If the legs are all out to one side, chances are it's expired. Either way, I stay loaded. I approach from the back of the animal, and I make sure I'm not blocking the path for escape. I used to touch the eye with the barrel of the rifle. Recently, I've switched to kicking the top of the head. If I am delivering a finisher, I get back at least 10 feet, and more if practical.
If memory serves, my last one that required a finisher was a decade ago. A big 6 point buck fell took one a little too far back, and bedded less than 50 yards from where I hit him. I was just approaching to touch the eyeball when he reared up and tried to trot off. He got another round of .308 at less than 10 yards that took out his heart, lungs, and guts. Yuck. Lesson learned.
Squirrels? We've had a few re-animate. I kick 'em around a little bit, before attempting to pick them up. Then they usually go into a plastic grocery bag before going in the game bag.
