Welcome Hawk23! Keep calling the permit office. You are likely to get a cancellation if you keep trying.
If that doesn't work, day hike permits for Saturday may still be available Saturday morning at the Visitor Center (1 mile S of Lone Pine, take the turn toward Death Valley). Overnight no-shows would come available at 11 a.m. at the same place.
If you're up for the long drive, you can head straight up to Horseshoe Meadows. Or if you head over the 50 and south on the 395, and need to stop, I have used various trailhead parking lots to throw down my sleeping bag on a Friday night. (You won't need a tent.) Just get up at dawn and start driving again.
It is likely you can find a spot at Horseshoe Meadows if you need to camp Saturday night. But you can probably get that overnight permit at 11 a.m. on Saturday for the Whitney trail.
