Mary,
Know first that less expensive will not mean less magical!
I am a huge fan of going in October, and although it is a bit more expensive than August, I think it pays off in getting to do more, and spending less time in line. But to answer your question, August typically has lower rates, but larger crowds. Staying to the first three weeks of the month (as you indicated you would be doing) will help, as the last week is very crowded.
From there, make sure you are looking at either a
Value resort hotel, or perhaps considering the
Fort Wilderness Cabins. While the latter are technically Moderate accommodations, they can take a larger family, and can be a much better deal. Part of the reason why is you get a full kitchen and outdoor grill, allowing you to have more meals in your "room" and saving money.
If you are not going to go that route, make sure you consider the
dining plans. They have always saved me money.
I am glad to answer any follow-up questions! Just visit my bio page to ask.
Have a magical vacation!