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Yes, you can export them, but not directly from the New Outlook app, as the New Outlook (the redesigned app) is still lacking in features when it comes to exporting.
The new version of Microsoft Outlook does not support exporting emails to a .pst file, nor does it allow users to download all emails from a folder such as Sent.
The best thing to do right now is to export your Sent emails from Outlook on the web (outlook.live.com or outlook.office.com). You may open Sent items, select emails (checkbox at top -> “Select all”), click PRINT, and there is an option to 'Save as PDF'
This saves emails in batches as PDFs to your desktop
Note: Does not export all at once, so you might have to run multiple batches.