FaceTime Will Not Activate

I resolved the issue by returning to basics.

I unpacked my old cell phone that I had thankfully not mailed off yet and called my provider, ATT.

A nice attendant made a switch from old cell phone to new cell phone somehow and the line went blank.

Then I could make and receive phone calls, but it took another day for FaceTime and iMessage to work.

Residual effects are that both FaceTime and iMessage still say they are unable to activate, even though they are both on and both work.

If I turn off iMessage and turn it back on, it says it is unable to activate.

If I turn off FaceTime and turn it back on, it not only says unable to activate, but the dialog box keeps repeating.

I can live with the residuals, I suppose. But as a mathematician, I want to resolve the issue totally and see the nice words "Activated!"

Thanks to razmee209 for answering my question!