K.W.
With a big family, you'll probably steer clear of downtown condos. You'll be looking at one of the outlying neighborhoods, probably in the City of Seattle proper. We live in one of the northern neighborhoods, Ballard. It has reasonably good access to public transit to get downtown (although the commute might be 30 to 40 minutes) and pretty good public schools. The neighborhoods all have different personalities.
Most of the northend neighborhoods are similar with good transit and schools (Wallingford, Phinney Ridge, Maple Leaf, Laurelhurst, Roosevelt, Bryant, View Ridge). I'm not as familiar with the southern neighborhoods (Georgetown, Columbia City, Beacon Hill). There are also a number of cool east-Seattle neighborhoods (Capital Hill, Madison, Madrona, Mt. Baker), but I'm less familiar with them. Check out West Seattle, as well, although the commute becomes a hassle when the bridge is out.
If you choose to locate outside of the City, Northshore School District has good schools. Your husband should still be able to catch a bus downtown pretty easily, although you might have to drop him off at a park-n-ride near I-5.
Eastside of Lake Washington (Redmond, Bellevue, Kirkland, Issaquah, etc.) is pretty nice and probably have good schools, but they are quite suburban. Not quite as neighborhood-y or walkable as some of the Seattle neighborhoods.
If you want to have a single car, I'd recommend picking an in-city neighborhood.
Good luck. Atlanta to Washington is quite a change. It's around 50 degrees cooler here today than most of the country and probably won't break 90 degrees this summer (defined as last half of July, August, and first half of September) more than a couple times.
For general info on children's activities, check into the Red Tricycle (Seattle) and Seattle's Child websites. That will give you an idea of recreational options, at least.