Ok, I read through every reply. I know some people are more thick skinned about this but the thought of shooting a dog just makes me sick to my stomach. I can understand how other people would feel differently and I won't judge but I would just like to offer my opinion and I hope that you read it and really consider all of your options.
First off, I believe this is your first time shooting a dog (horses are different from dogs). Most people here are saying that with one shot you will kill the dog instantly etc. but I am also reading other opinions saying that there is a possibility something else will happen and you won't kill the dog with one shot. Since I know nothing about guns and whatnot I can't say what I think will happen and I would think that since you have never had experience doing this, there is likely a possibility that something *could* go wrong and you don't kill the dog in one shot. If that happens the dog is suffering horribly and it is certainly not a humane way to end its life. On the other hand, if you take it to a vet or even a humane society, they have had experience doing this and know what they are doing and the dog will be able to end its life in the most humane way possible.
Personally, I feel that the dog has done nothing wrong and while it is really none of your responsibility, you are willing to end its life. I dont' think much more work is needed to take it to a humane society where they will do it for free. If you are worried about costs or whatever please pm me or post where you live and I will do some research online or call places to see where you can drop it off for free. PLEASE PLEASE do this, I don't mind looking this stuff up at all. There is the possibility that the dog will not have to die and will get fixed up. Some people are posting who would want a dog like this but I know people who adopt pets specifically like this because they love animals. My sister did this with her own cats. If there is any chance that this dog can live a better, happier life I would take it. Don't you think it deserves that much? And personally, I think taking it to a humane society will be less work than taking it to the woods and killing it. I still fail to see the reason why that route would be best or more humane. Again, if it's cost, I will research because I know places that will take the dog free of charge, if it's time, I think you will actually spend less time than if u did it yourself, and if it's because it's not your responsibility, then don't kill the dog anyway (it's not your responsibility!).
Lastly, I find it weird that you are posting about it on this board. You are obviously very steadfast in what you believe and even when others offer their opinions you listen politely but it's obvious you seem to have already made up your mind. What was the point of posting this? Maybe you have doubts? As reasons stated above, I find no reason that you, yourself, should have to kill the dog.