We're also going next October - maybe we'll see you there
I know you said you're not planning to rent a car, but have you looked at vacation rentals in the area? You can get a 3-bedroom house for $100-$150/night, ten minutes from Disney's front gates - which is actually closer than some of the Disney properties are to some of the parks, since Disney is so spread out. (Disney hotels "right outside the park" may be right outside Epcot and a twenty-minute ride to the Magic Kingdom, for example.) That also gives you your own kitchen so you have the choice of going back to your room and having a quiet meal, if the kids need down time. It does involve having a car, though.
I've found several resources for "good places to nap at Disney" - so even though it's nice to be close by, it's not like you have to skip naptime if you don't stay at a Disney hotel
(I'll be watching this thread with interest, though! I was at the Swan & Dolphin for a conference two years ago, but I haven't been to Disney itself for years.)