Please provide a few more details about what you mean of headers and messages being mixed up; about what you are seeing.
As a first step for troubleshooting this, select each of the mailboxes that are involved, and rebuild the mailbox. (Mail.app > Mailbox > Rebuild) See if that clears the issued.
If the rebuild fails, try a wholly new user; create a completely new login user, log in under that user, configure the mail account, and see if the weirdness still exists.
Thank you for your response MrHoffman. Herewith some details.
I imported my old user identity using migration assistant after setting up the new computer. This creates a new user account. Because their are no ways to connect the computers physically I used a time machine backup. It took a few hours for the 290 Gb to transfer. Then I noticed in mail that headers were being combined with completely random message content, some going back years. When I logged back into the original account I created, mail worked perfectly, as in my other iOS devices. I had an issue with garage band where loops no longer existed, but that was soon fixed by an App Store update that replaced them.
So, I decided to transfer the files with my wireless network, using the second account I'd created to migrate to the original account, (which is what you do if you want to put your files on the initial set up account when you make the mistake of not importing files during the initial set up). The transfer has been going 8 & 1/2 hours now and says 36/37 hours to go.
Obviously, I should have deleted all the files off the second account despite there being enough room on the 750gb HD before starting this. Or is there something else more, failing with my new computer, which hardly feels new anymore?
Apple support will be back online tomorrow morning, so I shall ask them. Any further advice, comments will be appreciated.
Ok; I'm not sure what you're doing. 36 hours is rather long. Seems like a new migration. Not what I intended.
Here's what I intended: from the newly-migrated and apparently-corrupt environment, create a new user, not related to any existing user, nor any migration-created user, or any other user for that matter. That is, use > System Preferences > Users and Groups, authenticate yourself by clicking on the padlock, and then click the + and create a wholly new user. Then log in under that user and establish the mail access.
36 hours? I'm wondering if there's an error or an exceedingly slow network here? Or a really, really slow disk? Or a sick backup? (WiFi isn't the path I'd usually choose, either.)
Failing the attempted second migration, I'd try a different tactic. Does your existing (old) system work? If so, I'd bypass the backup and connect an external (scratch) USB disk drive to the (old) sstem and then boot and use Disk Utility booted from the installer DVD disk or boot and use Disk Utility from the recovery partition or booted from a recovery partition created on some other external storage (details here vary by the OS X version and what hardware you have), and perform a full-disk backup of your original internal disk to (scratch) external storage. (Make sure you get the source and target disks chosen correctly here; copying the wrong way — from the scratch disk to your existing disk — will clobber your data!) In esssence, this will clone your existing boot disk. Then dismount the (formerly-scratch) external disk, transfer it over to the new system, and use it as the source of the migration, by performing a fresh OS X installation on the new system.
Target Disk Mode is also sometimes an option for accessing the disk for a migration, but that requires the right cable, and requires systems that have the same external connection; newer MacBook Pro systems use Thunderbolt for this, and older systems tend to use FireWire. And I'm guessing you don't have compatible hardware.
The details here can and do vary by your OS X versions and your particular Mac systems — if you'll identify the specific models and hardware, somebody might be able to better tailor the above (fairly generic) sequence to your particular configuration.