The risk isn't that someone will hack Amazon and get your payment details from there. It's that someone you tell will tell other people. If you avoid Facebook, avoid any promotion or conventions that will need real names, and don't tell people you know, the chances are pretty small of someone finding out. It's possible, but not likely.

Where most people fall down is they feel they must tell someone. You can't assume that friends and family will understand why it might be an issue.