This is a very tough question. When I was a lot younger I had a similar question regarding dating. Let me transpose the advice from there what will likely have some added value for this as well. I have friends who are like you, I also see what they do (not much). It pains me to see it since I was in a similar situation once in my life. Now I'm still weird and odd but I'm also social! :D And people seem to like that.

A couple of tips on finding truth in the social arena:

1. You have to find the truth by experimenting yourself. Set social experiments up deliberately and in a controlled environment [1].

2. Psychologists are mostly wrong due to the replication crisis. I didn't know this at the time, I've suffered the consequences I'm overfitted to detect human biases. I found that I know how people work much better than any psychology book (I did a bachelors in it). I also tested/experimented a lot more than any psychologist because I don't need to publish papers.

3. Self help books are about as wrong as psychology text books. My tip: go for the great classics (e.g. Dale Carnegie), ignore the rest unless you know that that person has a very similar profile like you.

4. When you experiment be ethical but err a little bit on the side for choosing for yourself. Chances are that you're too careful anyway. Slight transgressions are fine as long as you learn from them and rectify your mistakes. If you can't make mistakes then you're not in a place to learn anyway. My worst transgression was saying outrageous opening lines and looking at the effects of them [2].

5. Find books via HN just use the search bar or some aggregated data analysis on what books HN uses. That'll be a good application of 3.

With these tips you can find truth: ignore most books, test things yourself, do take the books for people who were like you (that's not an easy tip), err on the side of making mistakes. Personally, I haven't found an easier way and I learned this over 10+ years.

A couple of tips on dealing with anxiety:

1. Try to find core positive emotions that are natural to you. Mine are (in order): curiosity, fantasy/imagining things and playfulness (playfulness is already tricky). Identify it, frame everything like that. Curiosity goes really well with finding truth and experimenting. "How does this work?" is a question I often asked and tested.

2. Learn meditation, also helps in boosting emotional intelligence. I can write a book about it but I'll recommend you one instead. Search Inside Yourself from Chade-Meng Tan. Best book I know on the topic (I read a lot of them).

One tip on politics itself:

1. I don't know where I read it but it stuck. Social skills and political skills are different. There are people with good social skills who are not good politically. The reason is that political games are about groups not individuals. Learn how to divide and conquer (i.e. talk to multiple people 1 on 1 and push your ideas through that you are convinced about and think are good for the company).

On finding coaches:

1. Coaches are very hard to find. I've had several of them. The one that worked best for me was the one that showcased and demonstrated what was actually possible by doing it himself. All looked really social and good. But you need someone who's able to demonstrate and give real-time live feedback even during the conversation (in a covert way via text for example for obvious reason, or in ear also helps though I never tried that).

On using advice:

1. I am a sponge and would be too easily influenced by advice. A couple of questions you need to ask yourself for taking advice: (1) does the advice make sense to you? If it doesn't then why not (possibly do a couple of Google searches). If it still doesn't then leave that particular advice as on hold. Don't discard it but put it on the backlog and don't use it.

On dealing with people you tell that you're doing this:

1. If people are weirded out that you're methodical and as scientific as possible about this then discard that opinion. It's tough for other people to know what you go through since they never had this problem themselves. Even good empaths may not be able to empathize with you (though some obviously do, they are good empaths after all).

[1] i.e. not work but with strangers or if there can't go too much wrong then in your work environment.

[2] Spoiler alert: almost all my predictions were off, you can say some ridiculous stuff and have it do something other than completely mop you over the floor, friends who struggle with this don't believe it. Then I show them and they still don't believe it. Then I show them 5 to 10 times and they might consider believing it some day. Please experiment yourself.