Unable to verify account name or password in Mail OSX Sierra

Do a backup, using either Time Machine or a cloning program, to ensure files/data can be recovered. Two backups are better than one.

Try setting up another admin user account to see if the same problem continues. If Back-to-My Mac is selected in System Preferences, the Guest account will not work. The intent is to see if it is specific to one account or a system wide problem. This account can be deleted later.

Isolating an issue by using another user account

If the problem is still there, try booting into the Safe Mode using your normal account. Disconnect all peripherals except those needed for the test. Shut down the computer and then power it back up after waiting 10 seconds. Immediately after hearing the startup chime, hold down the shift key and continue to hold it until the gray Apple icon and a progress bar appear. The boot up is significantly slower than normal. This will reset some caches, forces a directory check, and disables all startup and login items, among other things. When you reboot normally, the initial reboot may be slower than normal. If the system operates normally, there may be 3rd party applications which are causing a problem. Try deleting/disabling the third party applications after a restart by using the application un-installer or by using the developer’s uninstall instructions. For each disable/delete, you will need to restart if you don't do them all at once.

Safe Mode - About El Capitan
also Sierra

Safe Mode Startup – El Capitan. also Sierra

If it works in the Safe Mode, try running this program when booted normally and then copy and paste the output in a reply. The program was created by Etresoft, a frequent contributor. Please use copy and paste as screen shots can be hard to read. On the screen with Options, please open Options and check the bottom 2 boxes before running. Click “Share Report” button in the toolbar, select “Copy to Clipboard” and then paste into a reply. This will show what is running on your computer. No personal information is shown.

Etrecheck – System Information

From system library delete the folder caches , and in user library delete the folders Caches and Cookies to the desktop , restart the machine and then enter the credentials , if it fails you have to clean the HD and reinstall the OS ( make sure back up the data ) .

Have you tried deleting the password from Keychain Access?

Read these articles : If Mail on your Mac keeps asking for your password - Apple Support

: Use Mail Connection Doctor - Apple Support

It has nothing to do with safe mode or cache cleaning from user library > cache folder .

The issue is with your system keychain , do you have your iCloud verification code in your records , the method that I suggested was applied many times in my system as I only concentrated for iCloud verification code that has major role and secondary are all passwords .

When new keychain is created the old is deactivated and the passwords have to be filled up manually ( by entering in internet accounts and configuring web mail and I cloud account ) , also the base station password in airport express .

Lastly the machine settings go to default and the home page is to be set up again Safari for Mac: Set your homepage in Safari

You can read the article Keychain for Mac: If you’re asked frequently to unlock your keychain

Try this article it will help .

Some more articles are there , read them to get more info .

macOS Sierra: Set up iCloud Keychain

Get help with iCloud Keychain - Apple Support

Frequently asked questions about iCloud Keychain - Apple Support

If you're asked to create a new iCloud Keychain security code - Apple Support

If you enter your iCloud Security Code incorrectly too many times - Apple Support

How to test an issue in another user account on your Mac - Apple Support

You are welcome. Try resetting the keychain again. Sometimes, deleting the account and adding it back works.

I told you not enter open mail and its preferences , but in system preferences > internet accounts .

I had some different method to set up keychain by entering in recovery mode then resetting the keychain in Mac OS Sierra , but still the iCloud verification code is required .

My personal suggestion is do not try to solve the issue by your own unless you are highly expert or have done it before , otherwise you have to install the OS again , you have posted the question few days back .

Consult apple support they will do it in a very short time .

Dear ,

Nlesh ,

When the Mac user set up a new iCloud Keychain account a 4 digit iCloud verification code is created and if you have saved in your diary or a safe place it will always help if some issues are in iCloud Keychain ( mean to say it is very easy to configure ) .

For every user the code is different , thats what I asked .

You are little bit confused ( please don't take it in another way ) , the two Mac have different preferences , don't configure the two Macs for the time being you can call in apple support via phone and consult senior advisors , they are expert and will set up your iCloud Keychain easily .

Best wishes !!

I have done various experiments in my machine few years back regarding keychain , in the starting couldn't sort out and have to install the OS , now what I do deliberately corrupt the iCloud Keychain in one of my computer sometimes , but always successful with the help of I cloud verification code that I always keep in my mind and written safely .

Any ways , your issue is been sort out and that is good .

Best Wishes !!