A little bit of reflectivity helped me a lot. When I say something stupid (it could be something socially awkward, or in the same way something obviously factually incorrect), I now see it as my brain failing me, without identifying with the error (I am a more long-term reflective process running on the brain, not its individual responses; I'm the one who noticed the error and will correct it, not the one who made it, if it's not the error of not training my brain better). I then look at the error and its causes, figure out a better decision, and try to train my brain the skill of noticing that situation and recognizing the better way of understanding it faster. This both reduces the negative emotional impact of making an error (less identification with its cause), and benefits the learning dynamic (more time is spent on solving the puzzle of how to do better, and not on blaming myself).

(There is of course a danger of falling into "not my responsibility, I don't care" mode. But your brain is still your responsibility, even if it's not the same thing as you. And teaching it important skills, such as generating more optimal thoughts immediately and not only on reflection, and not generating some wrong thoughts without needing to notice their wrongness on reflection, makes you stronger.)