She never hesitates to give herself a few seconds to ponder the question and give an answer, and she comes off as being highly reflective, thoughtful, and intelligent.

Here's an example of her taking a thoughtful pause: https://youtu.be/MaGAY_uMmcQ?t=1222 She does it throughout the video, but this particular example is a notable one.

So you have two choices, if this is a live presentation:

1) Get back to a healthy place right there, quickly. There are a variety of techniques for this which will take some practice. It can be as simple as taking some deep breaths. Or more complex, such as reminding yourself that their questions are not attacks, and answering them can open dialogue that takes you all to a better product.

2) Decide that you have hit your limit, and gracefully end that part of the presentation. I recommend honesty in these cases. Say that you did not expect the questions, and you would love to answer them, but are feeling that the pressure of the surprise questions is throwing off your ability to answer them well at that moment. Ask for everyone to allow you to finish the presentation, and send questions to you, which you will answer when you have some space to put in the proper effort to writing up the answers.

If this is not live, and instead is async written communications, you are free to step away for a time and come back to it when you are ready.

I honestly recommend a brief stint in therapy to learn some CBT skills. This is exactly what they are meant for - to help you identify when anxiety or other problems are stopping you from succeeding in your daily life, and give you tools to work with in those situations. You can learn these skills fairly quickly in a few therapy sessions, so it really is more like a few weeks of training, not formal "therapy".

Emotion-management is the key challenge here. Fear is the mind-killer which reduces your working memory. Your listening process was OOM-killed.

Specific suggestions:

1. Repeat the question back for the benefit of the audience.

2. Pause Believe in your right to claim time for yourself. If you're worried people are impatient, you can allay that worry by saying something like "Hmmm... Thank you. That is a GOOD question."

3. Say "I don't have the answer right now. That sounds worth looking into though...". Possibly follow it up with "I've also been curious about..." to take them on a journey with you.

4. Say "I had not prioritized that. Hmmm....".

You might think these all risk the possibility that you'll sound like you don't know what you are talking about. That is true--people being judgemental is indeed a risk. Ultimately the deeper solution here is to find a form of coaching to help you build courage.

I think I've heard Scott Adams describe the practice on a podcast, the first step to understanding your opponent is to ask them to summarize what they think you both disagree on.