Snow is melting fast now. (It's 100 F in Fresno as I write this, and Fresno is 50 miles west of Whitney.)
Reports are that many people are using the trail on the switchbacks. You will probably find some snow going over Kearsarge Pass, and likely over Forrester Pass, too. But if you don't do the passes extremely early or late in the day, the snow should be soft enough to kick steps wherever necessary.
I know I wouldn't carry axe or crampons. But extra balance offered with hiking poles might be helpful.