Welcome, lilschnetz!
June 30 is pretty early, so the heavy snow this year will most likely leave quite a bit for you to navigate, especially in the 99 switchbacks area above Trail Camp.
You can be sure there will be a well-traveled boot track through the area. What we don't know is whether you will be able to stay on the switchbacks, or if everyone will still be going straight up through the snow field just west of them. (Need to find Bob R's picture!)
Either way, if there is still snow, you will need to be careful about the slippery conditions. Early morning and later in the afternoon, the snow can ice up and become dangerous. Please be aware of that, and be careful.
Regarding the altitude, if you spend several nights just prior to your hike at elevations like Whitney Portal (>8000) or Horseshoe Meadows (10,000'), you will enjoy your hike more and be much more likely to succeed. Most of the people here in Cali also live between 0 and 500' elevation, too, so their altitude experiences are similar to yours.