Dear K.,
Well, at least you know that he trusts you more and feels closer to you so that he can be himself. This is what I think, it is a disorder, not an illness, and it is inherited. Just like you mentioned.
So, if I were you, I would see a counselor and explain the situation to him or her, then use the advice you are given. Whether it is to have your son see a counselor, or whether to find a way to get him into a 'safe' activity. You can begin thinking of 'safe' places where he is not required to interact very much with other people, like a library. I have always found libraries soothing. Then begin talking with your son, so that he does not feel threatened, or afraid that he will be forced, about how you two can work together to help him get over this fear. He will cooperate eventually, and the two of you can think of places that you could go together that would get him accustomed to being in the world.
You are a good mother, for goodness sake. Stop thinking that you are not. I have worked with thousands of parents, well, maybe 2 thousand, anyway we are all good parents, some of us need a big more support, but we are not bad. There is a saying that "all mothers need a mother", that doesn't necessarily mean that you need to talk more with your own mother. But, someone that is a bit older than you are, and you feel bonded to. You can talk things over with her, or him, a mother can be a man too, to help you feel secure in the ways that you devise to help your son.
My sister and I have a touch of agouraphobia - that is the name of this disorder, it means "fear of the market place". Anywhere outside your home can be uncomfortable to you. I don't know if I spelled it right or not. I have known a lot of people through my life that have this condition. People can learn how to conquer this fear, but they cannot be forced.
Good Luck, and go for it, but do not talk to Dad about it. He will not believe you and will give you a hard time. Men are different than we are, you know. C. N.