Now test these two things. When you are sending email as some non-problematic shared mailbox. Do you see an email in the Sent Item of the shared mailbox as well as in your Personal-Sent Items as well. What do you see if you send email from a problematic shared mailbox, do you still see the user(actual sender) copy. Run the Get-MailboxSentItemsConfiguration against the working shared mailboxes and let me know if the settings are same. Check if your Outlook is configured to be running in Cached Mode. If yes try switching to Online and see if still the issue re-occurs. Is DelegateSentItemsStyle registry key present or set with a value.Key: HKEY_CURRENT_USER\Software\Microsoft\Office\14.0\Outlook\PreferencesName: DelegateSentItemsStyleAlso on the problematic shared mailbox, change the settings as below and let me know the results.
Set-MailboxSentItemsConfiguration "Customer Support Feedback" -SendAsItemsCopiedTo Sender