Excellent plan, Bryan. Checking the last 3 years' Unused Whitney Permits pages, there have always been nearly 20 walk-in permits available, even on weekends. July 4 is when day hike permits get all used up.

As for hiking with someone, just start out on the trail, and talk with people you meet. You won't be alone, and others may be happy to include you in their groups. You will find friendly folks on the trail.