Greetings!
We appreciate your post in the Apple Support Community. I'm really sorry to hear about your frustration. It sounds like you're in a bit of a catch-22 situation. If you cannot remember your security questions and the support chat suggested using a new account, maybe you could try accessing your account from another device or browser where you're not logged in. This way, you won't need to log out from your phone. Hope this helps!
For English speakers, we have a dedicated community right here: discussions.apple.com. Since your message is in English, we warmly invite you to join us in that community.
Thank you!