You have a lot in your hands now, you cannot expect to make things works differently in one day.

You better re-prioritize: what are the most urgent things to be done?
Then think about yourself, your new baby and I am absolutely sure that your husband and you can teach your kids, however young they are, to do simple house chores. Any help will do, right?

Don't be afraid to ask for help even to people you don't know well: you explain the situation and you'll say that as soon as this difficult moment passes, you'll be more than glad to give back whatever they could give you.

As for the relationship with your Mum: seems like you are still in her loop, thinking the way she does and worrying about the things she worries about.
I guess it is time that YOU set the new rules of the relationship: you send out present with card but if you don't feel like going, you just don't do it and that's all.

Why not imagining that she is sick, which she is to some extent, and cancel all those thoughts of bad words and bad feelings. If your mother should act again in a way you don't like you can always pretend something to do, not being well or something and just leave her where she is, without any more reasons why and without strong words.

riotgirrl advice to seek help with a therapist is most sensible, because you need to revise all your behaviors toward your Mum and change them with new ones. Maybe it's too much to do all alone.

But we are always here