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Hi David,
From your problem description, when you opened .onepkg files which you exported from OneNote 2016 client, some sections missed. Based on my test in OneNote 2016 1909 version, the notebook can be exported properly without any unexpected issue.
As you mentioned that all the problematic notebook has the same error message about “This notebook may not be syncing correctly”, I think it might be caused due to the incomplete sync of certain sections in the notebooks. Hence, when you export the notebook, some sections are missing.
I suggest you can go to your OneDrive website which all the notebooks you created under this account should be saved in your OneDrive and open these notebooks in OneNote 2016 client, make sure the notebook has been synced correctly. You can go to File > Info > View Sync Status to check whether all the notebooks have been synced without any issue. After the notebooks have been synced successfully, you can click File > Export > Notebook > Select OneNote Package format >Export. Then the whole notebook should be export successfully.
Best regards,
Dihao