Oh good heavens - 4 weeks, my love, since giving birth! I am sorry, for some reason, I'd assumed we were talking about one toddler here. You are doing way too much, thanks to your Dh's expectations. Be very careful - if you make it the norm already that you do everything home and baby related, while he puts in his demands, that's a very difficult status quo to break further down the line.
Baby groups with a 4-week old? If you are enjoying getting out and meeting other mums then that's a brilliant way to pass some time each day. But, don't be pressuring yourself to go if you don't feel like it - your tiny little baby will not be fussed one way or another.
As for cooking and housework, if you're anything like me, it'll be taking up all your time just to get a washing done and do a bit of washing up as and when you get a chance. Everything else can wait, seriously. Four weeks in, you're just getting to know your little baby and find your way through each day. If I remember correctly, it's a good three months before you start getting into the swing of any kind of routine.
I'm a bit pissed off on your behalf that he sees fit to bugger off for the best part of every Saturday. What's he even doing for that time? He should be mucking in, giving you a bit of a break - or doing a bit of batch cooking himself, given his exacting standards of cuisine that he seems to be expecting.
You two need to talk, and divvy up the responsibilities here. You are looking after a tiny newborn, and he needs to be looking after you. Months down the line, when you get into your routine, and he's out working, you can then look after each other - that's teamwork.