Charter email iphone, Mac stop the sync

It sounds like to me the Charter email is IMAP, where as the Bellsouth was POP. What you are describing is how IMAP mail works, by syncing all mail between all devices. POP Mail does not do that because it is designed to be used on a single client, however you can access it from others depending on your settings.

If all Charter provides is IMAP, you may not have a method to do this.

Creating a mail account on the iPhone does not go through Apple for any verification. What you may need to do is verify the actual POP settings for Charter mail and then prepare the create the account on the phone. You can start it as a new account, but I suggest removing the Charter account you currently have on the phone. Then go to Settings, Mail contacts calendars, then Add account, select Other, select Add mail account, fill out the required entires and make sure you select POP instead of IMAP. If it has trouble with verification, it could have something to do with SSL, and that is on the Charter side as well.

See how that works.