I am also high on the anxiety scale and am a mother with a very close bond with my daughter. I can't tell you a sure-fire fix, but I can pass on some things that helped me.
One thing I realized, when I instantly went into anxiety mode over something that upset my child, or when she was sick, is that when I went into that mode I was focusing on myself, not really focusing on my daughter's needs. My anxiety and physical panic manifestations were because I was subconsciously prepping myself for something bad... fight or flight if you will. One day, out the blue, I realized that as a parent my most important job was to raise my child to the best of my ability. So when a situation cropped up, I really concentrated on what my daughter's needs were at that time. Does she need advice or just a sounding board? Would it be helpful if I offered to help her brainstorm some possible solutions? Does she need to to just lay in my arms until she feels better? I tried to really focus on her body language, actions, emotions etc so I could get the best idea on how to help her through her situation. If an anxious thought entered my head I would acknowledge it as if answering the front door and telling a salesman that it was not a good time to visit, I was busy, and could they come back later?
I found that when I put all my conscious effort into giving my daughter the best coaching I could, I was too absorbed in helping her to be displaying signs of anxiety. Then later, when I was alone, I would allow my anxious feelings and worries to be heard. I would mentally trot them out... "What if my daughter dies", "What if she develops poor self esteem", "What if..." And I would remind myself of rational answers for the worries. "She may die, and I would be devastated. But I would slowly go on living and never forget her", "If I see that she is developing a poor self image I will seek professional help for her", etc etc.
I can't say that my anxiety is any less, and I still worry. But most important, it does not have the power over me that it once did. When I get the pounding heart, breathlessness and lightheadedness I remember that it is only a biological response to an overactive mind. I try to observe it as a scientist would, just observing its effects on my body until the wave slowly passes (and it ALWAYS passes). Doing this keeps me from being "sucked into the vortex" of anxiety.
I am guessing that when the Buddha talked about non-attachment for lay people, he didn't mean that we shouldn't love our children. More that we should not grasp at our love for our children.