Q: I changed my Apple ID, and now I can't log into my new one for messages or FaceTime?

So I followed these instructions exactly as stated: Sign in with a different Apple ID on your iPhone, iPad, or iPod touch - Apple Support

I was able to successfully change my Apple ID in iCloud settings, iTunes & App Store settings, and GameCenter settings.  However, my iPhone 6 does not let me change my Apple ID for iMessage or FaceTime.

For iMessage, the instructions say to "Tap your Apple ID" under the "Send & Receive" tab.  However, my Apple ID is not there. I tapped "Use your Apple ID for iMessage," but then a window pops up that prompts me to enter the password for my old Apple ID... which, obviously, doesn't work.  A similar window appears under my FaceTime settings.

Is there another way to sign out of my old one?

Thanks!