congratulations on reaching day 11.
Constant hunger depends on what you mean by hunger. The hunger that most people refer to on a fast or even while eating has been described as "toxic hunger" or even the "all-gone-sensation" and "emptiness" within the abdominal area. This is not genuine hunger of course.
Handling food and preparing meals for your two school-aged boys can also be a stimulating influence and prompt your feelings or perception of hunger as well.
Exercise is also a known stimulator of "appetite" despite ketosis which will normally suppress it.
What concerns me is your mention of "chronic overfeeding" and "using food as a drug to self-medicate". This is a sign of an emotional/psychological disturbance as in "comfort eating" perhaps?
Dry skin can have numerous causes such as hormone changes in women as they get older, or even malnutrition; dry air; previous use of medications/drugs; hypothyroidism or low levels of thyroid hormone can reduce the amount of oils produced by your skin as well, so it isn't possible to pinpoint the exact cause. Insufficient water-drinking can also be a cause but unlikely from what you have said.
Water fasting though will resolve most of the above, especially metabolic disorders with the exception of malnutrition (highly unlikely).
Personally I would persevere with your fast, but try and get a little more rest; activity will exhaust your food reserves more quickly (and I am not just referring here to fat-stores)
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