Images have an older (correct) timestamp within the Photos app but exported files show the current date and time I have about 300+ photos taken in 2012 and it shows so in the Photos app. However, exporting it as original to a folder completely replaces that date with 2016 July. Not sure why, but even trying to extract EXIF data does not show the original 2012 timestamp.. So running EXIFTOOL to modify the dates would not work here, as the EXIF data with original creation date is not there. 

Any idea how can I force extract photos with the relevant timestamp shown in the Photos app?

For example, the following shows in Finder : 



But in Photos, I get this : 



And using REGEX.INFO, there are no date values in the EXIF :
At least with Photos v3.0, it does export the photos with the current date and time (which is undesirable) but the photos still have the original (correct) timestamp inside the EXIF data. That sounds like it may differ from your experience, so go ahead and check with Photos v3.0

If when you do export your photos again they still retain the correct EXIF timestamp but show the wrong Created timestamp on the file, then try using "Better Finder Attributes" app to first copy EXIF date to creation date, then to copy the creation date to the modification date.
(I have no affiliation with the "Better Finder Attributes" app, I'm simply a user of it).