Hi, April! I don't know how many people are in your party, but I went ahead to check the
Disney Dining Reservations system online. For a party of four on several of the days you are coming, I found availability at lunch and dinner at the Garden Grill at Epcot and Tusker House in Disney's Animal Kingdom. I was also able to see two dinner reservations on different nights available at the Crystal Palace in the Magic Kingdom.
It really is a matter of continually checking availability, and I have seen reservations open up literally within hours of our wanting to dine at a particular restaurant. People do have to cancel at the last minute for a variety of reasons, and you never know when you might luck into something.
If you did mistakenly make the dining plan purchase, as in you did not intend to include this when you booked your reservations, I would call again and ask for a supervisor. I'm not sure what they can do so close to your trip, but you should have time to make changes to your package if you need to.
Good luck!