When sending from a shared mailbox, it saves to your own sent, not the shared mbox.
Run this to fix it. It will save to sent on the shared mailbox.

Poweshell cmd "connect-exchangeonline"
Set-Mailbox -Identity "Mailboxname" -MessageCopyForSendOnBehalfEnabled $true -MessageCopyForSentAsEnabled $true
For the entire tenant - All shared mailboxes.
get-mailbox -RecipientTypeDetails SharedMailbox | set-mailbox -MessageCopyForSentAsEnabled $True