All the above are good. Here's more info...
Horseshoe Meadows pretty much always has camping spots available. They technically allow single-night only, but I don't think the rules are enforced. Here's an easy hike there: Trail Peak.
Since you have several days, you wouldn't be disappointed by heading up toward White Mountain. Grandview
is a campground around 8500, and puts you close to the 10,000' Bristlecone Pine forests, which would give you a fine contrast to the Sierra forests. Or you could drive all the way to 12,000' end of the road and walk up toward White Mountain
. (You could even bag the summit as long as it is early in your acclimation process, so you can rest before Whitney). Just remember to take all the water you would need to the White Mountains -- they're dry. You could even camp overnight at the road end -- no specific camp sites, but many people do that.
(A note on that White Mtn map: There's an 8-mile section where the road is not shown. Zoom out or switch to satellite view to see the road.