The Sierra Palisades region might fit the bill for you. It is often called the most alpine region in the Sierra with 6 summits over 14K, none of them with any route easier than class 2 and most of them class 4 or 5. It has several snow filled couloirs that become alpine ice climbs in the summer, and has the largest glacier in the area. It has some of the biggest multipitch rock climbs in the Sierra on Temple Crag. You could set up base camp in Sam Mack Meadow or the Palisade cirque and start knocking off routes, or if you wanted a bigger adventure you could spend all six days doing a high traverse of the peaks. The Whitney region is beautiful but very crowded and permits can be hard to come by. While the access trails to the Palisades are popular you will see far fewer people above treeline than you will in the Whitney region.
Get a copy of Peter Croft's "The Good, the Great and the Awesome" and start checking out some of the options!
Temple Crag, on the approach to the Palisade basin, with some of the longest rock routes in CA:
Palisades traverse takes you across this ridge, as seen from Mt. Winchell's summit:
North Palisade, as seen from the West:
Some of the alpine ice couloirs are visible here: