I don't have any pearls of wisdom about dealing with an anxious child, but I remember being one.Extremely oversensitive, terrified of change at school, terrified of getting things wrong, very fixed about winning or losing or losing face, very solitary in playground, very black and white thinking about various issues, such as whether someone was going to sit next to me or not. Academic in other ways. Fear of risky physical activities, climbing trees, horse riding.
What helped me, and made me into long term a less anxious child, with a friendship group, and happy enough at secondary was a) books, b) craft c)lots of dancing and nature based activities -woods and water d) pets (non judgemental) e) playing with younger and older children (I found my peer group quite challenging) f)singing g)non competitive games - in fact as little sport as possible, in comparison to walking, dancing and just being outside/swimming. All games which required winning or losing or skills made me very anxious, I was slow at processing or dyspraxic possibly.
I also think that I may have suffered a vitamin d deficiency (I lived in a northern clime, Ireland) and there was a run of grey summers in the 60.s which was corrected when I spent more time in the sun between 8 and 11 years old (moved to South East UK then)
HTH. It is awful being an anxious child, I remember to this day refusing school because I dreaded lunch hour so much, and frightened to tell anyone why I felt so bad. Physical feelings associated with fear and anxiety. My parents tried hard not to put too much pressure on me, having siblings helped ironically, and an outdoor lifestyle as a family. But I think they found me very exasperating to deal with, as they were naturally physically brave and energetic/fearless, rather than introverted. So I think adapting family life can help a lot.
My anxiety was much improved in teenage years and beyond. Although occasionally flares.