You're not dealing with a possum. If you were you wouldn't need to shoot it at all. I haven't come across any that are terribly fast. For a few bucks you can make an animal control noose with PVC and 3/16 cable. Don't be a moron and shoot things you can't clearly see. And even if you see it, for the love of god don't use a pellet gun... Yes maybe you can land a shot through the eye but possums and raccoons can be the antichrist of slipping away gently. A rabbit may get KTFO from a pellet gun, but try that on a raccon the same size. I've had some run off after 15+ .22 rounds.
PS. Did you eat these birds or just like killing and wasting for the hell of it?