After a lot of trial and error, I found the solution and am writing it up as much for myself as for anyone else who runs into the same issue. This time I was not going to reinstall the OSX - I’d had this issue years ago and had solved it then but could not remember what I did - just remembered it was simple but not obvious. On rebooting it I was back to square one when the OSX asked me to reenter my password for Google services - I immediately checked Mail and sure enough none of the accounts were connecting - same ‘ Unable to verify account name or password’ issue yet again. That was until I shut down the Mac a few days later. I was able to get emails and life was back to happy. Several hours later (rural broadband…) I had the Mac back up and running and all seemed to be ok.

I got onto Apples very responsive and helpful help desk, who in turn asked me to do safe restarts and other things but nothing would change so they asked me to reinstall High Sierra. I checked online webmail for a couple and the password worked just great so it wasn’t that. However, none of the passwords would verify. I have quite a few email accounts for different side projects and luckily I have the password all stored in 1password. When I opened Apple Mail all my email accounts failed to connect - all were asking for the password. I was not expecting this so didn’t have my 1Password available on my iPhone to help generate a new one so I muddled through and thought I would change that again later. During the install a new security check came up - apparently, you cannot have the same password for your Mac log-in and your iCloud log-in. Recently I upgraded to High Sierra - the Apple OSX released in August 2017 and all appeared to go well with the upgrade - it certainly looks fresh. Changed your keychain password recently? It may have wiped your passwords in Internet accounts I solved it with a fix that will save you time. Have you been unable to verify account name or password, or changed your keychain password and had issues with Apple mail and services such as Linkedin, Google, Flickr, Vimeo? I have had this situation. How to solve the Apple mail Unable to verify account name or password issue

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