But that mailbox appears as
Sent in Mail, not as
Sent Messages. The fact that you talk about
Sent Messages is what makes me believe Mail is displaying such a mailbox somewhere in the mailboxes list with a generic folder icon, hence
not using it for storing sent messages when using this Mac.
In the Finder, go to
~/Library/Mail/IMAP-username@mailserver/. With that folder open, do
Edit > Select All (⌘A), then
Edit > Copy (⌘C), and paste it in your reply to this post to let me see the names of the files and folders it contains -- you may disguise any mailbox names you wish to keep private.
If there is more that one
Sent-named folder there, which is the one that you would expect Mail to use for storing sent messages? Which of the
*.imapmbox folders within the "IMAP-" account folder in the Finder appear in Mail with generic folder icons somewhere in the mailboxes list?
Also, what are your settings in
Preferences > Accounts > Mailbox Behaviors > Sent?
Note: For those not familiarized with the ~/ notation, it refers to the user's home folder, i.e. ~/Library is the Library folder within the user's home folder.