Destination Expert
for Sequoia and Kings Canyon National Park
You could try to fit in Sequoia/Kings Canyon. This all depends on how much your boys like the outdoors. I've read some people here say that one should do Sequoia first because otherwise it's "disappointing" after Yosemite, but kids don't see the world through scenery-seeking eyes, so much. As a kid, my cousins, friends and I all loved Sequoia best, because there were so many places you could really romp and play and pretend to be a pioneer. Fewer people, for one thing.
Crescent Meadow in Sequoia is a place where a kid can find a stick to whittle his own walking stick, cross a meadow on a giant log, find a tree he can hide in, make a makeshift bow and arrow (bring string) and parents don't have to fear falling into rivers, waterfalls or off granite domes. They can fish (if you're into that), they can wade in the Lodgepole River, they can wait up to see if bears come.
The Lodgepole area of Sequoia has a sandy beach, one of the view in the Sierras that's okay to go in (it's pretty clearly marked where you aren't supposed to go in, and pretty common sensical too). There are horseback riding excursions, etc., and a granite dome to climb (not a Half Dome scary thing, but Moro Rock - something almost anyone can do).
The quiet beauty of Sequoia is its attraction. Also, there's more wildlife, since there are fewer people. You are likely to see bears in their natural habitat (no one has ever been killed by a California bear - and the only people harmed by one have either tried to literally pry food from its jaws or, worse, gotten between a mother and her cubs and attempted to interact with the cubs - they are used to people and do not want to interact with you). Yes, Yosemite has bears too. If you're camping, you have way more choices about where to stay in Yosemite - reserved sites still fill up fast, but there's quite a bit of first-come, first-serve camping.
There's lots of mountain driving on your itinerary already, though. You know what I'd do? I'd rent an SUV, buy a cheap tent (or a nice one), grab some sleeping bags and have a go at real camping. Actually - with kids that age, I'd buy two smallish tents and have them put theirs up (great fun). Sit by the campfire, really listen to the outdoors.
It takes a day to drive from Yosemite to Sequoia, because of all the windy roads.
At the end of your trip, all your camping gear should fit into one big duffel you buy at Fisherman's Wharf!
If you need advice on how to get gear in SF (or its outskirts), PM me or post on one of the National Parks boards (Yosemite people are very helpful, there's not a lot of us Seqouia people).