I do a lot of dry camping in the Sierras (Plumas, Tahoe, ElDorado NFs) and it could be hard to find campsites on the weekends. They fill up pretty fast usually (and the later the campgrounds open, the less likely a spot will be available because of the shorter season).
Check out the national forest website for non-reservable campgrounds. Those will be your best bet as you will be able to arrive during the week before others grab the sites.
With that, I did a quick search on some of my favorite campgrounds and with the exception of the Ice House area in ElDorado NF, none have notices of late openings. So, it might work out for you.
Since you'll be coming in at I-80 and going out around Vegas/Needles, you might go south on 395 from Tahoe and hit the Lee Vining and Mammoth areas. Check www.parks.ca.gov for info on Bodie Historical Park (they are currently accessing damages). The Mono Lake area is also nice. And that is just east of Yosemite, so you could hop over easily.
