I try update my Apple ID settings and it keeps coming up with verification failed

Some other 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.      

Receiving the message you received is common. But, I have not seen a definitive solution for it. I suggest you use my search link in trying to solve your problem: verification failed problem connecting to server. You will see a number of links that are, or may be, related to your specific problem.  And, if you want to contact Apple Support to deal with your problem, see:

Get a call from Apple support to help solve your problem using this

link https://getsupport.apple.com/?caller=kbase&PGF=PGF63005&category_id=SC0245&symptom_id=23362

Apple ID Support https://support.apple.com/apple-id

Apple Support Twitter https://twitter.com/AppleSupport

Apple ID account security for country-specific and

region-specific help https://support.apple.com/en-us/HT204169

Get Support: We’re here to help https://getsupport.apple.com/. Go through

the options to find the best one.