Are you sure this is true? My timestamps seem to update just fine.
Note, in FAT32 and NTFS, date created is different from date modified.
My json exports also sync fine on Dropbox.
Sometimes, you have to be careful with the export path.
You might change the export path before exporting but not save it again.
That'll make it forget where you exported it last and the next time you load the file and then export, you might be exporting stuff in the wrong place, making you think you overwrote the file you wanted (eg, the file in dropbox) when you actually didn't.