You want to export some photos as files from iPhoto, but the dates seem to be reset to the time when you do the export, not the time when the photos were taken. You want to keep the correct dates and times (which you might have also previously fixed using “Adjust Date and Time…”).
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Select all the files you want to export. For easier operation you might want to “flag” them first, and then “Select All” in the “Flagged” folder.
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Apply again “Photos → Adjust Date and Time…”, making sure to select “Modify original files” so that the Exif data on the actual files is updated. You might need to shift the dates of photos by one second up or down so that iPhoto allows you to apply the change.
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Select “File → Export…”, with kind “JPEG”, quality “Maximum”, include title, keywords, and location (if you wish), and set the size to “Full Size”. I’ve also found file name “Sequential” with prefix “Photo” and no subfolders to be the most convenient. Click “Export”, create and select a new folder where to place your photos.
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The files will have correct date and time in their Exif data, but not in the file creation date (as shown e.g. in Finder). To fix this download jhead, place it on a convenient location, and make it executable:
$ chmod +x jhead
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Finally use the jhead command line tool to fix the date and time of all files:
$ jhead -ft *.jpg