I have exactly the same problem.
I upgraded my iPhone 6 to iOS 9.3.3 just last Tuesday (didn't reset/backup) and noticed the problem the next day. I also upgraded our iPad Mini soon after that but all built-in dictionaries are still available there.
I contacted Apple Support via Twitter and they want to run a diagnostic test as soon as I get to backup my iPhone (which I have yet to do as I am too busy and Internet connection here is sooo slooow). I just hope it won't go as far as having to reset my Phone and restore it from backup (which I'm guessing won't help based on others' replies).