My personal reccomendation is Canaveral National Seashore. The reasons that I like Canaveral are: not crowded AT ALL, natural sand dunes still intact, wildlife, and you can see all the cool NASA stuff in the distance from the beach. It costs $5 per car to get into the seashore area. While there you can also visit Merritt Island National Wildlife Refuge which has many nature trails (walking and driving) and a manatee viewing area. The two parks are side by side, and there is a very nice visitor's center that covers both.
It takes about 45 min to an hour to get there from the Disney area. We always take 417 to 528, exit to get to I95 to Titusville. From there you can pretty much follow the signs to Canaveral, but I think it's the second exit once you are on I95.
A bit of warning: this is a rustic beach which means that there is no running water. There are bathrooms of the type that they come clean the tank out periodically, but no showers, sinks, or water fountains. Also, nowhere to get food in the immediate area, so pack a picnic lunch and some drinks. I know this type of beach isn't for everyone and may not be your style, but the time of year that you're visiting is the perfect time for trail walking and seeing wildlife.
Let me know if you have any questions!