I reserved a permit for Shepherd Pass a few days ago at recreation.gov. It appears that “Overnight exiting Mt. Whitney” permits are available. I’m sure that Steve is right that the lottery is unavailable during the shutdown.

Admir: You’ve already done Grand Teton, so you should have an idea of how well you tolerate altitude. Tolerating altitude and being fit are not the same thing. There are several threads on acclimatization, AMS, Diamox, etc., on this site. Two bits of experience I will share: First, the best sleeping altitude is the highest altitude at which you don’t develop symptoms. Lone Pine Campground is better than Lone Pine, but Whitney Portal and Horseshoe Meadow are even better. Second, current advice is to take Diamox before you ascend. I wouldn’t do that unless you have a history of AMS or you have no time to acclimatize. Diamox still works if you wait until you need it. And if you felt OK on Grand Teton, you can probably forget about Diamox.