But are there OVAs?
Of course.
On called simply _The Future Diary OVA_ predates the series and is basically a <we should totally have an anime> proposal covering a few scenes from the first few episodes. Nothing noteworthy.
You want _The Future Diary: Redial_
The Third World Gasai is having déjà vu and memories that don't make sense.
She goes on a quest to the spirit realm and retrieves the fragment of Second World Gasai's memories, then breaks through the dimensional barrier to be with Amano again.