Some places I have luck getting into last minute:
Magic Kingdome: Plaza, Tony's and LTT for lunch
Studios: Prime Time, Sci Fi
Epcot: Nine Dragons, San Angel, Garden Grill,
Animal Kingom: Used to not have full service so a lot of people don't automatically think of table service eating with that park. You can try for Tusker House or Yak and Yeti and see what happens. Rainforest Cafe at AK takes walk ups and if you join their Safari Club for fifteen bucks you get priority seating and a discount.
The resorts have lots of restaurants. I have gotten in last minute at Kona Cafe at the Poly as well as the Grand Floridian Cafe. The moderate resorts also have full service restaurants, all three of which are quite good. They are lovely resorts and fun to check out, if you have a car this especially might be a good idea. The Turf Club at Saratoga Springs and Olivia's at Old Key West almost always have availabilty and can be gotten to via boat from Downtown Disney.
If you are on the
Disney Dining Plan for sure call now and get what you can. You don't want to run the risk of having table service credits you cannot use. Even if you aren't on the
DDP I suggest you call and just see what you can find. I am going in two weeks and was able to get Sci Fi, Crystal Palace and the Hoop Dee Do Review just a few weeks ago. And those were just the ones I tried for, there were in all liklilood lots more available. I am also going in late May into early June and just last weekend booked a few more ADR's with no problem.
I really think that there is always something available somewhere on Disney property, especially at off times. Don't give up without trying now and again once you get there (if you don't get exactly what you want now, people do cancel and last minute adr's do open up).