As to the fear around improvising, it's the same as the fear of dancing freestyle on an empty floor. It won't happen if you are worried about being judged for it. When you can turn off that fear of judgement, it will be easy.
Interesting. I was told a number of times that I appear to be hugely confident (especially on the dance floor), while in reality I feel anything but confidence. I guess I learned to cover that up. Although some, I guess, well-meaning relatives said that I look hard to approach.