Sorry you did not get the permits. I didn't get the camping spots I wanted this year twice and was quite unhappy. I have not tried to join a group, but I was successful a few years ago with the last minute permits. The rangers are always there, so no going around them.
That said, do you have any flexibility in your dates? Because frankly August is not the nicest season to hike half dome, and it will be super crowded. It's not nice because it's very hot. Later in September will be much more pleasant and the crowds will have thinned out, so your odds are better. Or try and go earlier in the season. There will be crowds, but it won't be as hot.
A comment to all others who say don't worry about half dome, go climb elsewhere, while in theory I agree with you, at this point in time, for many folks, even myself, half dome is like a rite of passage. It's one of those things you can talk about and complain about the crowds, IF you've done it. I did it just before I turned 40 a few years ago to prove to myself that I still could do it, and while it took me much longer than it did when I was 18, I felt good that I still had it in me to do it.
By the way, if you do go, be sure to take the time on the return trip to soak your feet in the Merced.
If you have those days off in Yosemite and end up not getting the half-dome lottery, others have suggested some great hikes. I would go for one of the Tuolumne meadows hikes, as by then, most of the mosquitoes will be dead, and it will be a nice time to be up there. Mt Dana is short, but strenuous, and gives you out-of-this-world views. You can claim you climbed the second highest peak in the park.
Saddlebag lake and all the lakes beyond it are an easy walk, again, with some incredible scenery.