As an INFJ, so NF, I see sensitivty as part a gift part something to overcome rather than accept. It's a double-edged sword but you've got to overcome the black side of the double-edged sword. Getting hurt too easily does you no good most of the time, being overly self-conscious does you no good most of the time. Although I agree that being self-conscious is a good thing when kept in check or balanced with a backbone, it's about manners and being polite. The people who are completly unaware of themselves are not exactly role models.
Beside getting hurt easily, other problems are not being able to stand up for yourself and always assuming the worst. You have to be able to take care of yourself as a human being, there's plenty of nice people in the world, but not all people are nice people, and you need to be able to stand up for yourself and confront or deal with not so nice people. You need to have some dignity and a sense of pride so that you will protect yourself or your worth and fight back. Being an aggressor is not good, you're a toxic individual who harms others, but being weak is also not good, nobody is impressed or awed by a rabbit in the same way you are by a tiger. Even if you're not an aggressor, for you shoudn't be, you should be able to bite, as psychologist Jordan Peterson puts it, for others will bite, and you need to be able to fight back.
There's a difference between being an aggressor and assertive. It's the difference between the one who attacks and the one who protects himself or the difference between the one who initiates the conflict and the one who just fights back. It's as the Japanese say: It's better to be a warrior in a garden than a gardener in a war. I know this is very abstract but hopefully you get the point. It's not a one specific scenario in mind but more of a general thing.
And the last one is always assuming the worst or overthinking. In that case, look for alternative explanations. Yes, sometimes things may be the worst, so your fear is founded. But other times things may not be the worst and it's equally likely that you're acting based on the worst case scenario when you shouldn't. That in itself is a mistake. Not all real life situations are like defusing a bomb where making a mistake is fatal. Most mistakes are just good source of learning so that you eventually get more experienced, so don't be afraid of making mistakes when it's not so bad if you fail.
But that's not just about you fearing of making a mistake, but also about you evaluating the situation at hand, cosndering the likelyhood of the bad thing to happen and the good thing to happen. What if a relative doesn't call you and you think that they find you bothering and this is why they don't want to reply. Is it possible that they find you bothering? yes. But, and this is a big but, it is also possible that they simply forgot the phone and aren't paying attention to it right now, and you worry about the worst case scenario over nothing, when that worst case scenario is not even real. Just because you have a worst case scenario in mind, it doesn't mean that worst case scenario is also real. There's a saying: aim for the best but prepare for the worst. So prepare for the worst, but act as if you are hoping for the best until or unless further evidence proves you wrong.
Just because something is not your natural talent it doesn't mean you will never become good at it. It will never be your natural state, which is part of the gift for sensitivty has its perks, but it will be necessary and useful to use when you feel threatened or negative.