Questions like yours come up every year. I’ll try to help, but you’ll have to decide for yourself what you feel comfortable doing.
First, self-arrest with an axe can save your life, but it’s not rocket science. I would watch some videos and try to find someone who can show you how to do it somewhere near home. If you slip, you need to get into the proper self-arrest position quickly. Practice doing it until you can do it instinctively. Guide services typically make you slide headfirst, flip around, and stop. Practice that, too.
Second, many people want to skimp on footwear in the snow. “Why can’t I just put MICROspikes on my trail runners and go up the chute?” MICROspikes are for low-angle slopes. They are fine for parking lots and trails with minimal snow. They just provide a false sense of security on more challenging terrain. I would bring real crampons (ten-point or twelve-point) with front points unless I was sure that I wouldn’t need them. Boots are critical. Crampons can come off with disastrous results at the worst possible time. A stiff boot works much better. In fact, a stiff boot (by itself) often gives better traction than MICROspikes on trail runners. Grab the heel in one hand and the toe in the other. Try to twist (not bend) the sole of the boot. The stiffer the better. Take the money you would have spent on Shasta, and get the proper gear. You may not need it on Whitney, but you WILL need it when you get around to Shasta.
Finally, you might end up waltzing up Whitney in trail runners in mid-June. My wife and I didn’t bring any gear the first time we did it on June 22, 1988. Check the trip reports in early June.
