Sorry, another account from your organization is already signed in on this computer.

When you try to sign in to an Office 2013 app by using your Office 365 user ID and password.

This was rather frustrating as I was signing in with the same account, but had recently changed the users primary login email address from the *@* domain account. 

Here are the steps needed to get this resolved;

This method isn’t recommended by Microsoft, unfortunately I had tried everything else, the issue remained.
To fix this issue, you need to remove the existing user account and all connected services from from your Office 365 profile on the local computer, you will also need to clear cached credentials that may be on the computer. 

Step 1 Remove the user account from your Office 365 profile

  1. In the upper-right corner of any Office 365 app, click your name, and then click Switch Account.
  2. On the Accounts screen, click Sign out at the top.
  3. Locate the account that you want to remove, and then click Sign out.

Step 2 Remove connected services from your profile

  1. Go to File, and then click Account. You may have to open a blank document.
  2. Under Connected Services, remove all the services for the existing account.

Step 3 Clear cached credentials

  1. Edit the registry to remove cached credentials. To do this, follow these steps:
    1. Click Start, click Run, type regedit, and then click OK.
    2. In Registry Editor, locate the following registry subkey:HKEY_CURRENT_USER\Software\Microsoft\Office\16.0\Common\Identity\Identities
    3. Select the Office account that you want to delete, and then click Delete.
    4. In the Identity subkey, locate Profiles, right-click the same Office account that you deleted in step 3 of this procedure, and then click Delete.
    5. Exit Registry Editor.
  2. Remove the cached credentials in Credentials Manager. To do this, follow these steps:
    1. Open Control Panel, and then click Credentials Manager.
    2. Under Generic Credentials, locate the account that you want to remove, and then click Remove. (You will notice a Office 365 reference under each credential.)
  3. Log off, and then log on to the computer. (you may not have to do this if Office 365 is already closed and nothing is running in your system tray)