Shots of desperation? Why not wait a few seconds for a different opportunity? A deer with its head down as you describe doesn't sound to be in panic mode.
Are you the one in panic mode?
Ever spent 5,10,15 minutes watching a buck before even deciding to shoot...or letting it walk?
Yep.
Give them a second. When they raise the head, on a close shot, shoot 'em at the base of the skull where it meets the neck. If it's at distance, you have the a large drop zone from the top of head to the chest.
Neck shot = DRT, so it's one of my favorite shots.