The Sent Messages under On My Mac is the local mailbox created when you sent a messages without storing messages on the server. It got pushed out to On My Mac local mailbox section when you rechecked to store messages on the server. You can right click on it and delete it.
The problem is that for that account only, the IMAP sent mailbox seems to be hidden. I don't understand why it won't appear under the exchange folder when you uncheck storing messages on the server. This may be a problem with the account itself. For example, gmail has an option to not show mailboxes as per your choice. But if that were the case, it shouldn't show on other devices.
Let's try these things. Be sure to Quit Mail before doing this.
In the Finder, select Go, then Go to Folder. Paste this in the window:
~/Library/Preferences/com.apple.mail.plist
A finder window will open to that com.apple.mail.plist file. Move it to the trash. This will put your Mail application back to it's default settings. Open Mail and see if the Sent mailbox for that account appears.
If that doesn't work , quit Mail. You can select the com.apple.mail.plist file in the trash, right click on it and select to Put Back if you want your original settings back.
Once again, in the Finder, select Go, Go to Folder and paste this:
~/Library/Mail/V2/
This will open a window to the V2 folder in your Home Library. You will notice a folder for each account. Move only the folder for that specific account to the desktop.
Then open the MailData folder. Select the Envelope Index files and move them to the trash. Open Mail and hit Continue in the message box. This will reindex all messages. This could take a while if you have a lot of messages, so be patient.
Please make sure you have some kind of backup before doing anything. If you don't, then copy the V2 folder to the desktop or some other location so your messages and accounts can be put back if necessary.