When I give my Password for My Apple ID, I get "Verification Failed: An unknown error occurred."

Try to login to iCloud using your browser and see if it would work. In some cases, it is just a mistyped password. You can, of course, initiate a password reset/recovery also.

Some things you can try. Do you have access to another WiFi network where you can try? That has worked for several people.

Try a restart.

Restart your iPhone   *

Restart your iPad.   *

Try a forced restart. See the section towards the end of the article.

Reset Device. -