Can’t add email address to my Apple account

To change your current Apple Account to a new Primary Email Address, make sure you are following these steps:

Change your Apple Account primary email address - Apple Support

If that new email address is associated with another Apple Account, it cannot be used. If that email address is a Recovery Contact for a different Apple Account, you will have to remove it from that account and there may be a waiting period before it can be used as an Apple Account.

There are 2 different passwords and I am not sure which one you are referring to. A gmail address will have a password. This is to send/receive emails and if they changed it, then you would no longer be able to receive emails.

The Apple Account also has a password and in order to create an Apple Account, you have to enter the verification code sent to the gmail address used for the Apple Account. That would not be possible unless your gmail account has been compromised. If your gmail account was compromised, you would need to contact Google.

Had you previously set up an Apple Account with that gmail address? If you set up the Apple Account with two factor authentication, no one would be able to sign into the account even if they knew the password, unless they were able to enter the code sent to a Trusted Device (which is a device that is already signed into your account), or the Trusted Phone Number that was previously used to set up the Account. If you believe it was your Apple Account that was compromised, then follow the steps here:

If you think your Apple Account has been compromised - Apple Support

Since I am not sure what actually happened, it is difficult to provide a solution. In most cases when another person is able to take over an account and change its password, whether that is a gmail account or Apple Account, it is likely lost and you are better off starting over with a new account. There is no positive identification used when creating either account to prove it is yours, and the security of those accounts is the Password that only you should know and additional security measures such as Two Factor Authentication to prevent unauthorized access.