My 5 year old (non-CAH) also has extreme mood swings.  He gets upset over the most ridiculous things and can get completely out of control.  He cries, screams, flails his arms and legs, throws things, etc...   He also complains of headaches but I don't think it is the same as you described with your daughter.  They don't seem very severe and never last long.  Incidentally he also complains of every other kind of ache he can think of.  Other than these tantrums, he is a great kid, very well behaved. 

I was talking to my sister on the phone one day when he went into one of his "fits" and she said that her son did the same thing until he started kindergarten.  It was like he just needed some structure and stimulation.  So I am hoping that my son will be better in the Fall when school starts.  Like your daughter, my son never seems to have these tantrums when he is at pre-school.  I also think part of my son's problem may be that he doesn't get enough sleep.  He usually wakes up several times during the night.  We are trying to work on that.

So anyway, the problem may have nothing to do with CAH.  But unfortunately, I have no solutions....  Good luck!