I think your problem is that there I dont believe all the sit down locations are open yet and at the ones that are, they are spacing out reservations/limiting capacity. Just keep checking. A lot should open up by July.
Resorts are a great place to look for alternatives too. For instance if you cant find anything at Epcot, The Beach Club Resort, Boardwalk resort and Swan & Dolphin are a short walk away.
There are a few web sites
like this one where you put in your desired location and details and they will alert you when the slot opens up. You gotta be quick when you get the text b/c usually there are a bunch of people hounding that table.