There are two issues with this, I'd say. It sounds like you are overly bothered about appearances and what other people think if coping with ill-chosen comments on a night out, when people are drunk, is a question of saving and rescuing your partner. It does come across that you're protecting your joint image and that these superficial considerations are important to you. If your friends have been drinking too, then surely they are embarrassing themselves. This is what drink can do.
Too much alcohol is bad for the health; it sets a negative example to your child. I would try to persuade him to change his lifestyle - do it together, for this reason. Say to him that being over-chatty isn't a good style. Don't make it more than it is: it's an indiscretion and social faux pas, but potentially nothing to counter-weigh being a brilliant dad.
The other strand is talking about people in high places: these 'friends'. It's surprising that you don't know who all his friends are already - after all these years?? If you are finding out worrying things about him which you didn't know before - as drink is loosening his tongue, then that is more worrying. Anything to do with crime, or a fantasist personality, is something to be concerned about.
Generally, all people can have cringe-making habits, but, if you love them, it's usual to speak nicely about it and laugh it off. Character is more important. Being cool and socially adept is nothing in relation to the bigger challenges of life to do with physical and mental health - challenges, you, he and the baby may at some time face. Having a loyal, honourable partnership as support through any future troubles is very valuable. It's not always easy to be a single parent But the 'friends in high places' conversation suggests you need to be sure that there isn't danger there from something a lot more worrying than getting a bit verbose.