Young Lakes (via Glen Aulin trail,) Glen Aulin and Nelson Lake all have openings for July 25th. So if you send in your reservation form now you do have some choices. And there will always be walk up availability to some of the trailheads, you just can't predict exactly which ones.
Young Lakes is a truly spectacular high country destination, it would be my choice for a late July trip. 7 or so miles to the first Young Lake, the next two are just beyond. Stay here for two nights and you will have several interesting dayhike opportunities such as Ragged Peak, Mount Conness and Roosevelt Lake.
Glen Aulin will be past its waterfall glory by late July, so nice but not as spectacular.
You find Nelson Lake on the reservation application form under cross country trail heads. As others have said there is a trail to Elizabeth Lake, but after that is is use trails to Nelson Lake and pure cross country beyond, so you need to be confident in your cross country navigation skills.