"No one familiar with the eastern side of those parks?"
Drive up to Bishop, CA, go in over Paiute Pass down to the JMT, exit over Bishop Pass and either hitch or walk the few miles back to your car at the North Lake TH. This loop fits your time parameters and, depending on your hiking mileage/day would offer ample opportunities for side trips. It would definitely give you a feel for the Sierra. Another trip further south would be to turn west at Independence, CA, drive up to Onion Valley TH for Kearsarge Pass, hike over Kearsarge Pass down to the JMT, go south over Forester Pass and exit via Shepherd Pass. You would probably want to schedule a shuttle pickup there to take you from the Shepherd Pass TH back to Onion Valley, but it would not be very expensive, as the distance is not that much, perhaps 6-8 miles or so. This route also offers opportunities for great side trips into Wright Lakes Basin/Wallace Lake Basin, over to Lake South America, or even into the Upper Kern Basin, as the distances are short and the hiking easy. This area is quite different from the trip over Paiute Pass and personally I think it is by far the more beautiful of the two. It is also more strenuous, as you will be going over 13,000' to cross Forester Pass and Shepherd Pass is a rugged trail, even coming down. I find it worth the trouble. Opinions will vary on this point, of course. Hopefully this will be enough to get you started.