Apple ID suggestions ERROR “ Verification failed. Unable to connect to the server”

I can't guarantee a fix, but see:

Verification failed: There was an error connecting to apple id server https://discussions.apple.com/thread/7238348

Error Connecting Apple ID, Verification Failed. How-To Fix https://appletoolbox.com/error-connecting-apple-id-verification-failed-how-to-fix/

Fix: Verification failed ‘There was an error connecting to Apple ID Serverhttps://appuals.com/fix-verification-failed-there-was-an-error-connecting-to-apple-id-server/

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

Try a restart.

Restart your iPhone, iPad, or iPod touch         

Try a forced restart.

Reset Device.      

If necessary

Hard reset.        

Do a backup and try a factory reset. Then restore from your backup.

Restore to Factory setting.      

Backup - Restore your device from an iCloud or iTunes backup.      

Your WiFi is showing that it is not connected. Have you tried using WiFi to connect?

Troubleshooting Wi-Fi issues in IOS.