If you will have one down day to acclimate, then the best option for you is to drive up to Horseshoe Meadows (which is not too far from Whitney Portal) and hang out there for a day. You could also hike up to Cottonwood Pass which sits at 11,100ft. or you could hike up to Trail Peak which is at 11,600ft. Both are accessible from Horseshoe Meadows - Cottonwood/Trail Pass Trailhead. This will give your body a good day to get use to the altitude. But spend as much time as you can there - the more time at high elevation the better you will perform when hiking Whitney.
Good Luck!
