Mail: Unable to verify account name and password

There are other threads on this - most of which conclude you have to trash your exisitng account and then add it back in. That's all well and fine but I'm concerned my email history might not be merged into the new account. Some of these are business accounts and I can't afford having disjunct email records spread across multiple instances of accounts.

Oddly - Mail Connection Doctor reports - "Connection to server success; login not required."

Yet Internet Accounts reports - Can't connect to the account..."xyz". Of course all the credentials are correct.

I tested email and while it cannot SEND email from this (GoDaddy) server - receiving email works just fine. Not to mention, this account was running just fine some days ago and nothing has been changed anywhere by us.

I have 8 email accounts in Mail - ranging from FatCow and GoDaddy hosting to Gmail and Yahoo - SENDING from the actual server is/was messed up in about 1/2 of them since OS X 10.11.2 was installed. Now this - "Can't connect to the account" issue....

Any ideas?