My kids always have a blast at Disney Springs. They usually have live entertainment scattered around. My boys love the lego store, where they have big tables of legos that you can build with for free. There are also a few QS locations like Early of Sandwich and a pizza place that you might be able to get away with sharing. They also have a boat that runs from one side to the other that you can ride for free.
Also don't forget about the water light parade that runs through the various Magic Kingdom resorts. I believe the one at the Poly was at 8 or 8:30 p.m. when we were there this past July. It also hits up the Grand Floridian, Contemporary and Fort Wilderness at different times throughout the night. It isn't super exciting---but free and one of my favorites from growing up.
The only time I've noted the beach area being closed to visitors was during some days during the Christmas holidays (Christmas Day and maybe New Year's Eve?), never noticed it closed during Thanksgiving week, and haven't been to the World in the Fall.