This is spot on. OP, you need to read this over and over until it sinks in that you are making hasty decisions that are massively impacting both your life, and the lives of 3 children.
Did he even want your baby? If he's 4mths, then he was conceived 13mths ago. Making the twins about 2. And you say he only split with their mum when they were 2. How long were you actually together when you got pregnant? He wasn't planning a baby with 2 yr old twins, was he. In fact, so delighted with the immediate addition of another baby, he's had the snip to make sure it can never happen again. He was just too stupid to use protection, and has made sure he can never find himself in the position he has with you, again.
You don't see all this though, and it's honestly frightening. He's not your partner, he sleeps at yours to have his dinner made when he gets back from work, and sex. He pays you nothing. And you are convinced you are a little family, actually thinking you are moving in together.
You need to move on from this, the sooner the better. This is the most you are ever going to see from this man. When he finds someone with no children to move in with, you will barely see him at all. I know we all sound like doom mongers, but at a little bit older and wiser than 25, we have seen this happen 1000 times.
Raise your standards OP. Focus on you and your children. Stop clinging to a man who has no respect or interest for you or your child.