Overcoming anxiety can be a long process requiring a great deal of introspection. This introspection is generally more effective when led by a trained therapist (Cognitive Behavioural Therapy in particular for anxiety).
However, overcoming anxiety in specific situations can be done.
We must learn to :
recognise situations that make us anxious (as you have done in your question)
Learn tools to relax ourselves so that we can sufficiently distance ourselves from the anxiety to challenge our fears.
Challenge our fears, even if it is in our imagination that we challenge them.
You have already identified situations that you find anxiety provoking. So, once you recognise those occasions you can use techniques such recognising that your breathing has become shallow and slowing it down, taking deeper breaths, relaxing your shoulders, changing your posture. There are many such techniques that can be found online (if i find some good ones I'll edit and link here)
Once relaxed we can challenge our thoughts. When anxious situations can become distorted in our heads. For example someone anxious about germs may believe they will die from shaking someone's hand. This is an obvious distortion of what is likely to happen. So we recognise and challenge these beliefs. This can be done by looking at evidence for your anxious thoughts, and testing out the reality of negative predictions.
This does not only have to be done mentally, as this is a long process. It is possible to challenge these thoughts by conducting experiments. Observing others doing the same behaviour and seeing the evidence that nothing bad happens to them, before testing assumptions ourselves. We must weigh the pros and cons of avoiding the thing we fear and determine the realistic chances that what we're anxious about will actually happen.
From there we replace these negative unrealistic thoughts with more relalistic ones.
So. skipping the process:
Negative thought: People will judge me and think I am foolish
This is a distortion of being blown out of propotion.
Realistic replacement: People are more likely to understand and look for ways to be helpful as people like to be helpful. (evidence - the existence of stack overflow :P)
This is obviously difficult because you are replacing patterns of thought you have built up over your lifetime. However, if you can learn to recognise the triggers that create anxious moments and calm yourself then you can practice challenging them every time.
I am not a therapist This is best done with a therapist. I am aware there are many CBT self help courses online and a great amount of online documentation in this area. I cannot vouch for any of them. I get all my information second hand from my Significant Other who is a trained therapist. (giving you this information so you can evaluate this as a source information. I may get my SO to edit it for me)