Look at wawona, in the park not on the valley floor, close to south gate and often overlooked. In fact, what I have found, many folks who miss out on the 15th of the month reservation opening booked Wawona as a backup then cancel if they can get in to one of Pines CGs on the valley floor. You might just get lucky and pick up a cancelation.
I think May is the best month in the park.
We have taken our dog several times, although Yosemite is certainly not the most dog friendly place.
2007 Chevy Duramax LMM/Allison
2012 Jay Flight 29QBH
2012 Jay Flight 26BH (Sold)