we travel in January, May and September and have found varying patterns with walk-up availability. These are not the busiest times of year, of course...our January trips the parks were extremely quiet & empty. But our May trips have had crowd calendar days of 7s & 8s; September as 5s & 6s.
By far, the most availability was to be found at resort table service restaurants. Maya Grill in January was virtually empty at 6:00 on a Wednesday (and it's a beautiful relaxing restaurant with great food & excellent service.)
I suppose if you're wanting the most popular character meals in the parks at prime meal times it's much more difficult but if you're not there at the Most crowded times of year, and you're flexible about where & when you may be surprised at a number of choices.