I found that if you're saving the Word document as docx, you can edit the XML directly and remove the timestamps this way. Follow the procedure below if you want to do this:

1. Rename the file to zip so it becomes a ZIP archive and you can easily open it using the appropriate tools
2. Open the renamed file, it will now open as a ZIP archive
3. Open the word folder in the ZIP archive
4. Open and edit the document.xml file in a good text editor. For example use Notepad++
5. Search using the following regular expression:
w:date="[\d\W]*\w[\d\W]*\w"
and replace with nothing. This regular expression will match all XML
timestamp attributes and remove them from the document.xml file.
6. Update the ZIP archive with the modified document.xml file
7. Rename the ZIP archive back to docx sothat it becomes a Word document again