I was at Horseshoe Meadows last Labor day weekend and it was full and overflowing. We had to make new friends and share some space. It usually isn't full but a holiday weekend could be. I also think Rock Creek is a good option. Since you have a lot of time you should check out the bristlecone pine forest and that campground never fills up. The Grandview campground is at 8600 feet. There is no water and it's dry but you'll get to check out the bristlecone pines nearby which are the oldest trees in the world. You can drive to the end of the road and do some hiking up White mountain. I think the gate is well over 11000 feet where you can park. You can follow the dirt road all the way to the 14k summit.
