Remove "FULL NAME" from email address in Apple Mail

The "FULL NAME" field appears in my email address on messages that I send in Apple Mail on my Mini-Mac with Sierra OS. It appears in both the iCloud email account address and I also have a bellsouth.net email account in Apple Mail.

For example:

John Smith <example@bellsouth.net>

John Smith <example@icloud.com>

There must be a way to remove personally identifiable information (FULL NAME: John Smith) from the email address.

I have tried changing my name in Contacts. I have tried "Nicknames" in Contacts. Neither worked.

In Mail, under Preferences / Accounts in the "Email Address" field I have the following:

Under the BellSouth IMAP:

John Smith <example@bellsouth.net>

example <example@bellsouth.net>

Under the iCloud IMAP:

John Smith <example@icloud.com>

example <example@icloud.com>

In BellSouth IMAP I have example <example@bellsouth.net> selected and in iCloud IMAP I have

example <example@icloud.com selected. However, the From name on my emails are still

John Smith <example@bellsouth.net and John Smith <example@icloud.com>. I have tried deleting the "John Smith" entries in both the BellSouth IMAP and iCloud IMAP, but the John Smith still appears in the From email address.

Any suggestions as to how to remove this personally identifiable information would be appreciated.