I've noticed this same problem. I took mine into a local apple repair center and they kept it for a day to run some tests, then gave me a new pencil wrapped in plastic (no box). I tested it as soon as I got home today and.... same thing happens. It still begins to draw before the tip touches the screen. I've tested a couple different apple pencils and they also seem to barely do the same thing - beginning to draw while hovering 1mm above the screen. Sometimes it feels more like 2mm, and while the center of the tip may be 2mm away, I believe part of the curved edge is more like 1mm away.
Overall it is still a pretty amazing little stylus with the pressure and angle sensitivity, but I was really hoping it was something that could be resolved. I don't know if it is something that can be corrected with software, if so, I think the Apple pencil would be that much closer to flawless. Hopefully we see an improvement with the sensitivity sometime soon.