Logging out of old and unused Apple ID

I recently changed my e-mail to use as my Apple ID because my old e-mail got hacked and I cannot logged back in to that particular e-mail account. So, I changed my e-mail through my Apple Web Page (appleid.apple.com). Unfortunately, I forgot to logged out from my Apple ID on my devices first before I changed my e-mail through the web. And as of now, my devices are still using my old Apple ID which---of course--no longer exists. My devices kept telling me to log in to that nonexistent account, but I couldn't. So, how can I deal with this kind of problem? It would help me so much if anyone could answer this and help me. Thank you.