Are you using secondary authentication account in which three answers for respective questions , date of birth , rescue mail address and its password is added , for your old Apple ID account .
As you wrote - But I can't access the email to reset my password for that account.
Possibilities are the rescue mail address password is forgotten and the rescue mail address and Apple ID address both are same , and if you try to reset it using iforgot.apple.com the code goes to it but you can't open it .
So reset the password of rescue mail address from the web account ( the rescue mail address could be created with some mail services like gmail , yahoo ...etc ) .
Another method to reset the password .
If you click on - answer security questions - click on the right arrow .
Date of birth , verify your date of birth to continue > fill up the credentials > click on next and same pop up window will appear .
New password ….
Verify password ….
Fill up the credentials and click on reset password , the password is changed .
Once account is accessed change the rescue mail address .
Go to iforgot.apple.com enter only Apple ID and click on continue .
The rescue mail address is always added in the Apple ID account , sign in with Apple ID account , if it is not yet added create a web mail account say with gmail service .
Make sure this mail address and password should be totally different than your Apple ID address / password see the section of rescue mail address in this article https://support.apple.com/en-in/HT201356 in the same Mac open a new tab ( press command and T keys ) sign in with your gmail account here in the web mail account you will see this code , come to the old tab in which apple id account is already opened , fill up the code in the fields and click on done to save .
And you are trying to use a restricted method changing password from iCloud.com only and not using iforgot.apple.com using the same Mac , please clarify .
