Apple support contacted by someone pretending to be me
You should way the risks and ensure that if in fact someone is trying to take advantage of your account or even using social engineering to compromise your account, you minimize the damage that could happen.

• You should call Apple support again, perhaps using the same ticket number, and explain your situation yet again. Explain to them what you just explained to us. If you can convince them, they will likely get a senior support advisor to handle the issue. Perhaps you can negotiate some sort of verbal password with Apple support such that anyone calling them on your behalf would need to know the verbal password before Apple would make any critical account changes. You also need to be careful that the other person does not convince Apple that they are the real you and you are the imposter.

• In order to make any account changes via Apple support, the caller needs to know some information about the account other than just the email address. After all, an email address is fairly public information. You need to make sure these pieces of info are not obvious. For example, your birth date would be required, however this is also fairly easy to find info.
You therefore need to make sure your security questions are secure, and by that I mean the answers to your security questions are not easily discovered information. Note that the answers to security questions do not need to be accurate. You just need to be able to remember them.
I also recommend setting up two-factor authentication on your Apple ID, which eliminates security questions with amore secure unique one-time code.

All security-related settings may be found at appleid.apple.com.