During the "skillfulness" stage of acquiring a new skill, you are able to handle yourself well, but you still need to actively think about what you're doing.

That statement is true
Skillfulness is the third stage of acquiring a new skill, right after you finish the awareness and awkwardness stage.
In most cases, people entered this stage after making several trial and error when doing a certain act and started to develop a new sense on how to do the specific skill efficiently.