FaceTime call on my iPad: trouble signing in

I am trying to initiate a FaceTime call on my iPad2, as I have done numerous times before without being asked to sign in or anything like it.

FaceTime now asks me to sign in with my Apple ID and password. However, it does not recognize my sign in.

Assuming I forgot my password or did not update it, I clicked 'forgot my password', got the requisite email invitation to change it, and changed my password.

However, FaceTime is STILL unable to verify my email address and gives me a message "Unable to verify address. An email with a verification link has been set to the above email address". Despite doing this a number of times, NO EMAIL HAS BEEN SENT TO MY EMAIL ADDRESS.

Is this a scam to get my/our Apple ID passwords???

What is wrong?

How do I solve this?