In mid-late April 1984, a snow year not unlike this one, we were able to ski about 5 miles to the Mineral King end of the road. It had snowed a few days before, though, and most of this new snow was melted by the end of this weekend. With 4WDs, then, we might have driven to a mile or two from the end of the road.
It was solidly snow-covered well up to Farewell Gap, then. But that same year, on Memorial Day weekend, the hike to Vandever Mountain was mostly bare.
I found the last few hundred feet to the summit of Florence was full of large talus and I had to turn back due to hazard of falling into a snow hole (solo). I wouldn't know if you'd have to worry about this in a few weeks, yet.
You may wish to plan a contingency extra day for skiing Florence. Seems with the still short days, the 5,000' gain from the summer road's end would take some time, and it is pretty steep to ski, with avy hazard as well.