10-11 nights is plenty of time to see both Yosemite and Sequoia. If it was me, I'd probably try to do both. Bearpaw doesn't open until mid-June, but depending on trail conditions you might be able to hike up the High Sierra trail in May. Unfortunately May can be a tough month to predict what road and weather conditions will be like, but you should definitely be able to go wherever you like if you come later in the summer.

It really doesn't matter in what order you see things. Yosemite is a bit closer to San Francisco vs. Sequoia though. It takes about 3.5-4 hours to get from SF to Yosemite Valley, roughly 4 hours to drive from the Valley to Sequoia, and about 4-5 hours to drive from Sequoia back to SF. If it was up to me, I'd probably see things in that order, making a progression from big city to busy park to quiet park, but it's all a matter of personal preference.

It would be a good idea to post in the Yosemite forum if you have not already done so to get an idea of what trails you might be interested in, but there are many hikes in Sequoia that will show you what makes it such an amazing place. Personally I would recommend the Big Baldy trail, a hike up to the top of Moro Rock, and of course the trail to General Sherman. These latter two will probably be the busiest areas in the park as they are short trails and popular sights, but they are must-see attractions IMO. The Big Baldy hike should be right up your alley though. If you're feeling ambitious, you can also try the hike to Alta Peak, but that may depend on the snow situation as well. Hopefully this helps!