There is a meltwater tarn right at Shepherd's Pass. It is larger, deeper, and cleaner than the tarn at Whitney's Trail Camp. You should find good water this time of year, but it's been 5 years since I have been up there.
There is also a seasonal pool at the Tyndall-Polychrome Col. That is on the way to Williamson Bowl from Shepherd's Pass but that will likely be dry.
There are also lakes and campsites in the Williamson Bowl if you want to get closer to the beast itself.
As for snow, I went in June of 2013 and there was no snow that we couldn't bypass. I don't think you will encounter any this late in the year.
@jjoshuagregory (Twitter & Instagram) for landscape and mountain photo spamming...