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Unfortunately I don't know of any application to transfer only messages, but I recently was in the same situation and there's really two options: 1) clean up all the crap you don't want to backup, then backup to icloud or itunes and just restore to the new phone, and 2) use iExplorer (paid version) to download and save all messages in PDFs which are searchable, including the attachments which can be saved as files in subfolders. I chose the latter and think it's sufficient for my situation.
 
Deleting all the crap, and just leaving vanilla plus Messages, then backing up - now that is a good idea!
 
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