Can't sign out of Apple ID

Try first to optimize the range of the mobile network, see this Apple article:

If you see No Service or Searching on your iPhone or iPad

If you see No Service or Searching on your iPhone or iPad - Apple Support

Low Power Mode is On.

Charge the iPhone & wait until the battery is fully charged (100%).

The Low Power Mode will turned off automatically.

Try again.

See if anything here helps. 

https://www.imyfone.com/ios-data-erase/9-tips-to-fix-there-was-an-error-connecting-to-the-apple-id-server/   -

https://appletoolbox.com/error-connecting-apple-id-verification-failed-how-to-fix/.  -

https://macreports.com/verification-failed-there-was-an-error-connecting-to-the-apple-id-server-fix/.  -

https://appuals.com/fix-verification-failed-there-was-an-error-connecting-to-apple-id-server/.  -

https://errorcodespro.com/there-was-an-error-connecting-to-the-apple-id-server/.   -

Verification Failed" There was an error connecting to the Apple ID server -

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, 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.   *