Re White Mountain:
Apparently they are not doing an open house this year until September:
http://www.wmrs.edu/community/open%20house/default.htm. This has implications for the dirt road leading to Barcroft Gate. The lab used to do some work on the dirt road before each August open house, but I don't know what their plans are this year since there is no open house. So, if you do not have a high-clearance vehicle, you may want to call the White Mountain lab,and find out what the condition of the road is. (For reference, when we went up for the open house in 2010, we used a Volvo station wagon, and we were fine, though drove slowly.)
By the way, it was sleeping at the Barcroft Gate (elevation 11,700') the night before the hike that I discovered I need a small dose of Diamox at high altitude. Turns out I have altitude-related sleep apnea (also known as periodic breathing or Cheyne-Stokes) above 11,000 feet -- something I had never experienced at 10,000 feet. However, I gather this is not uncommon. Since I was carrying Diamox (although I had not planned to take any), I was able to solve the problem with a small dose on the spot, as periodic breathing is one of the altitude-related issues that Diamox can help address. I mention this as you may want to talk to your MD about obtaining Diamox, as Steve mentioned earlier.
Re going solo up Whitney:
I know you are getting a lot of advice to go solo. Recognizing that reasonable minds can differ on this issue too, I, personally, would not recommend to a newbie to high altitude to do Whitney solo as a first timer. A lot of folks on this board have done Whitney a number of times and/or have a lot of high altitude experience, and I think that may affect perspective. Two years out from my first high altitude experiences, on White and Whitney in 2010, the combination of fear and excitement is still fresh for me. I did exactly what you are doing (researched the mountain and asked lots of questions), and summiting Whitney (which capped off a 1.5 year weight loss effort) was one of the peak experiences of my life (pun intended!). Really fabulous. (And I did go with friends, all newbies like me.) I do recognize that people have different sensitivities to risk taking.