Re: Having it say 'in game content' but not being installed.
@TabbieTheWix Store worlds like Barnacle Bay and Dragon Valley need to be downloaded through the Sims 3 Store, not through the EA App. Please check both your purchase history (Store > My Store Account > Purchase History) and their product pages, in case the worlds show as owned in one location but not the other.
https://store.thesims3.com/barnacleBay.html?categoryId=12642
https://store.thesims3.com/dragonvalley.html?categoryId=12642
It looks like you also own Celtic Lands, the add-on for the gold version of Dragon Valley that may need to be downloaded separately as well. At least, it doesn't download alongside DV for me, so double-check the launcher.
If you see this issue with an expansion or stuff pack you've bought, like you have with Showtime, just clear the EA App's cache:
https://help.ea.com/en-us/help/faq/clear-cache-to-fix-problems-with-your-games/
You shouldn't have needed to purchase the world twice, and if you'd like a refund, please contact EA customer support.