hedwards: I can't blame you. Whether you made the right call or not, it just sucks. And I can't say that I have any suggestions about how to avoid the problem.
Personally, I probably wouldn't do contests, I'd probably do games and then decide whether or not there should be a prize afterwards. Probably cut down on most of the scamming if there's no way of knowing if there's going to be a prize other than the satisfaction of winning.
KobusSteenkamp: I'll keep it in mind. I actually wanted to make a couple of more giveaways.
I'm not sure there is a particularly good solution to the problem, but the rules can help a bit as can the way you handle it. People often times have minimum rep requirements and age of account rules. Give aways that favor many entries with no effort are going to be scammed more easily than giveaways that require effort. The more the giveaway focuses on actual activity, the harder it's going to be to scam.
I don't personally participate in giveaways on either end, but I think if you look at some of the game based giveaways and ask the people running them, you might get some good ideas.
I'd hate to see people give up on generosity, but the generosity with no work on the part of the recipients is a part of the problem. Kind of like feeding the birds.