he gets anxious in any kind of social situation except closest family and friends, and yet with those closest he can be very likeable, humorous, accepting and interesting.
Unless he has other ASD traits which he displays in non-social situations (which you haven't mentioned) I think it may be more likely that he has a social phobia.
I get very overwhelmed socialising in large groups, especially if I don't know the other people very well. A party is my idea of a living hell!
My ideal group size for socialising is 4 people or less. If I know the other 3 very well, I can handle adding a couple more people that I don't know well.
I also cope better when I know the environment where we're meeting. EG a pub or restaurant that I have been to at least once before, preferably with someone close to me. To sort of reconnoitre the layout!
I don't cope well with people that I don't know coming to my home. I see my home as my refuge and having strangers there makes me feel like I have no safe place to retreat to - whereas if I'm meeting someone out, I can tell myself "If it gets unbearable I can say I'm not feeling well and go home" (even though I have never had to do this - it makes me feel reassured that I can if necessary.)
Over the years I have built up a kind of avatar of myself that can do the socialising thing without coming across as too awkward - although sometimes I worry that I come across as a bit shallow instead! I call it "putting on my work face" and I basically pretend that the people I'm meeting are new colleagues and I have to speak to them and give them a good welcome. Basically I have to pretend that I am interested in their lives. (God that probably sounds really weird and awful...)
It's a very individual thing, but maybe your DH can adapt some of my coping techniques to help calm his anxiety?
You will have to accept yourself, though, that he's never going to become a smooth and polished socialiser! You will undoubtedly have more fun going to parties and large gatherings without him, and socialise with him as a couple only in small, familiar groups and settings.