When you have a situation of a user having full access and send as access on a shared mailbox and the user sends an email send-as or on behalf of a Shared Mailbox, the sent item will be saved in the user’s mailbox and not in the Shared Mailbox.
To fix this, there is not option in the GUI so far so you would need to connect via Poweshell as the Global Admin. Note: If you are using Multi-Factor Authentication use the App Password to login instead of the password.
Open PowerShell as Administrator and type the following
$Cred = Get-Credential
$Session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri https://ps.outlook.com/powershell -Credential $Cred -Authentication Basic -AllowRedirection
set-mailbox "<mailboxemailaddress>" -MessageCopyForSentAsEnabled $True
set-mailbox "<mailboxemailaddress>" -MessageCopyForSendOnBehalfEnabled $True
This will still save a copy in the user’s mailbox but will also save it in the Shared Mailbox