Is this what you used?
Want to make changes to your Apple ID? You can update your email, mailing address, or other account information. You can even change your Apple ID.Sign in to manage your Apple ID.
Password & Security, you can't see your Security answer, but can change/redo it. It does show your Birthdate.
Hello......... it certainly looks like the, or one of, the pages i used. When i started the process i was surprised to find we can't alter our email addresses on our profiles and there aren't any instructions on how to do it, i looked on the support site and found many others had encountered the same problem and i think we think its pretty dumb. It was from the answers that i found the link and was able to change the email address then it just fell apart after that. I hadn't changed my password, and wasn't planning on it, but time and again it rejected my password and informed me that as a matter of security my account had been locked and i was to click a link to deal with it but that didn't work either.
There were two choices offered, one is an email to verify, but it didn't turn up so i chose the 2nd option which was to answer security questions, the first question being about my birthday and it was rejected... about 3 times.. i might have typed in the wrong thing the first time but i was very careful after that and still it wouldn't work. I thought i was really stuck until i managed to change the password and things fell into place.
I have looked at the link you sent and my birthday is (correctly)there yet was rejected.
I have customised a question cos when i saw that my previous question was about my hometown i couldn't remember whether the answer was where i was born and grew up in Australia or where my hometown in NZ is ! Hopefully avoiding trouble at a later date.
Interestingly i read several posts from people that have had exactly the same problem with correct answers to security questions being rejected out of hand. Something can't be right. good news is that i now have broadband, the Apple site was SLOW on dial up and that would have made me even more furious.