There are a lot of options but they are not as well known.
-- FW cabins: already mentioned
-- 2 rooms at value or moderate: already mentioned
-- 2BR suite at DVC property: ideal if you can swing it. Quite expensive at rack rates but significant discounts are often available at Old Key West which has the largest suites anyway. Renting points can cut the expense almost in half -- see DVC rent/trade board.
-- If one of the six is under 3 you can actually stay in one room at the deluxe resorts that normally sleep 5 because they have a daybed in the room. (GF, CR, PR, BC, YC, BWI) A little crowded but nice!
-- Junior Suites at Wilderness Lodge, Swan/Dolphin, Contemporary. Again expensive at rack rates but maybe viable option if you can get major discount.
-- Doubletree Guest Suites near Downtown Disney: technically on property but not Disney-owned. A good value if you can get a reasonable rate. Large suites about 600 sq ft with two double beds and a sleeper couch. Similar to typical Embassy Suites but not as nicely furnished.
-- Sheraton Vistana Resort: just outside WDW on route 535. Timeshares but available for cash rental. Nice 2BR suites. Mixed reviews but definitely worth investigating.
-- and of course all sort of off-site options, including vacation homes for rental.
(No, our family is only 4! But I have complete sympathy for those with more.)