The thing about most baby-and-mother groups is that you don;t need fabulous social skills - most of the other mothers will be in the same state as you. That's why they can be such a relief. The ones that i went to often turned into consciousness raising sessions: "OMG! This is so hard! And there is so little awareness of how hard it is in wider society!"
There are two threads on mumsnet at the moment about how tough it can be. One is called something like "does motherhood affect your mental health" and the other one is in the Feminism section.
You really are not alone.
Totally agree with Dogs' suggestion about leaving your partner in charge and going off for at least a day-trip. You could go for a day trip to the park, with an alarm clock, and sleep if you can;t actually go away somewhere overnight.
I didn;t ever do that. With hindsight, I think it was a mistake not to. Really. I could have expressed milk, and just done it. I think it would have been good for my sense of self, been good for the bond between dh and the littlies, and been worth a thousand words of explaining to dh what my life (and needs) were like at that point.
It does all pass. It gets a lot easier, very quickly. It really does. But it is not your imagination that it is hard now.