Will I have to rebuy my games?

As long as you sign in with the same account you can download all your purchased games again, no problem. The same would also be true on a new PC.

You should check your payment through the option you chose when purchasing as if they took both payments you should be due a refund if you have the proof. I would get on to Xbox support over the phone or through chat as this should never happen.

You can check your transactions at http://commerce.microsoft.com to make sure the purchase went through. The game was also free to play at the same time it was onsale so if the purchase doesn't show on the billing
site, it was downloaded for free to play during that weekend.

Have you tried uninstalling and re-installing the game?

Unfortunately without the original account, you'll need to buy the games again.