We are in the same position, having just reached elite. We called Princess after our last cruise and they changed all future bookings to show elite (which it does on the website). On the app, I show elite, but DH shows blue! He called the medallion helpline and they said they would order his medallion as elite and change the app. The app still shows blue.
We pick up our medallions at the port as we are In Canada, so will have to wait and see. Fortunately pre-cruise I can access the booking to sort out our dining and will get priority boarding. We suspect we will get a blue and black medallion at the port. We will go to guest services when on board to get it changed.
And you should have priority boarding as platinum, get your status changed for the website and print your travel summary to take with you. That will show correct status.
For the dining reservations, maybe another member can chime in.