jbabs, a nice way to combine sights and acclimatization would be to climb Mt. Dana in Yosemite. It's the quickest, shortest route to get to 13,000 feet in the Eastern Sierra. It's a 3-mile hike from the Eastern entrance of the park on Tioga Rd., so six-miles round-trip, with a little over 3000 feet of gain. The beauty of doing this as an acclimatization hike is you can see how you perform up to 13K' without burning your legs out prior to your Whitney hike (White Mountain is a long hike). From the summit, the views east over Mono Lake and west to the interior of Yosemite are spectacular. No permit required, and you don't even have to pay the park entrance fee - just use the abundant roadside parking right outside the entrance station. The trailhead is directly behind the entrance station, to the south.
The bad news is it could still be a longish day if you're camping in the Lone Pine area (the Portal, Horseshoe Meadows). It's a 4-5 hour round-trip by car. You can complete the hike in 5 hours going at a slow, steady pace, maybe 6 if you linger for the scenery. You can cut that drive down and still sleep very high if you were to camp further north (Mosquito Flats or Onion Valley). Another option would be to lodge for the night in Mammoth Lakes at 8000 feet and be only 30 minutes away from the eastern Yosemite entrance station, where the trailhead is located.
You could even spend a second day seeing the sights in Yosemite Valley, if you're so inclined. Fair warning - the summer crowds in that tiny valley are ridiculous. You might as well be in Manhattan. Half Dome, El Cap, Glacier Point, all the falls - they're worth it at least once, though.
Big ditto on Steve and Wagga's warning about Diamox dosage. You take 500 mg, you'll be buzzing like a high-tension line. A 62.5 or 125 mg dose gets the job done without most of the awful side effects. A post-hike beer should taste right . . .
