I would say you are both b a bit u.
Unless you feel you need the support/social interaction of other mums at the moment, I wouldn't voter with going to m and b groups yet. 4 weeks is too early to understand what's going on and ime at that age they just want to be with mum and dad at home-getting to know they're surroundings and eventually getting a routine.
As there's a difference between tidy and dirty. Even with a newborn I/you can manage to put a wash on, Hoover, dishes,empty bins etc. and a meal from scratch takes 20 mins if you're canny about it.
However, it's all new to you and there is no mad rush. Bonding with your baby and getting into a routine is more important than anything else. It does get easier in time.
But your dp needs to lower is expectations of you and support you and pull his weight while you both adjust to parenthood.
Parenthood is not like anyone thinks it will be like. It's hard in the first couple of months.
If I were you, I'd sit down with dp now and go through the issues you both have and work at resolving them.
He cannot expect to have 8 hours to himself in his free time. Not at the moment while baby is so young. You should be doing things as a couple and a family together in your free time.
And you need to get into a routine where to are able to do more housework. (I'm aware ill get flamed) but if pure busy with baby and dp at work, who is going to do it?
Has to be teamwork.
And congratulations with your newborn.