This just made me really sad. You're 23 and you're living as his house-keeper.
I'm a SAHM. My husband works. He does all the cooking when he's here. He comes in and takes over the childcare. Not so that I can get on with housework, but so that I can put my feet up. He brings me a cup of tea first.
When he's here we share the work 50-50. If ever I say "sorry, I didn't manage to do a wash today" he says, "Why are you sorry? Neither did I!" And washing is a job that I said I would always do.
You do not have to put up with this. If you're worried no one else would ever want you... well, my ex-husband used to say that. I was a lot older than you when I left him (he did a hell of a lot more around the house than your dp), and had kids already. I am now happily married (something I never thought I would be)
It's up to you to decide. Do you want to live like this forever?
Talking is always a first step. Sit him down, say you've had enough. Tell him you've been thinking long and hard and that his lack of respect for you is shown in everything he does. Tell him you've been considering his offer to move out, and that it seems like a good idea with the way things are now.
Every time he leaves something for you to do, he's saying a silent, "Fuck you, you do it"
"Fuck you, you pick up my clothes"
"Fuck you, you wash the dishes"
"Fuck you, it's your job to cook for me"
He's a parent just like you, his job does not start and end with earning. You are contributing financially to the household by looking after the children that are equally his.
Have you thought about getting a job? It would open up the possibilities for you and give you some real independence.
This is not how it's supposed to be.