Depending on what permit you can get, Trail Fork Springs would be a bit closer to water than Anderson Flat, although it’s not too hard to hike down to the spring from Anderson Flat.
I think Anderson Flat is a little barren up on top of the ridge and like Trail Fork Springs Camp a little better. Shields Flat is more sheltered than Anderson Flat, but there’s no water at Shields (unless you melt snow).
Limber Pine Bench on your way in should have water at the zig-zag of the first major switchback above the camp.
High Meadow Springs will have water, but it’s quite a ways off the main trail along the divide. You have to drop down steeply to the camp area and then drop down even further to the meadowy area below the camp where the water is. Water flow at High Meadow is always reliable but is always low. There typically are no deep pools from which to gather water. You just have to hunt around in the meadow until you find a spot where you can gather water with relative ease.
If it were me, I’d stay at Trail Fork Springs, tank up with water there, and bypass High Meadow Springs.
Your next reliable water, except for snow, is at High Creek as you come down the Vivian Creek Trail.