Breakup? How did you do it?

Hi everyone... I’m 21 years old, engaged, and have 3 month old baby girl with my fiancé of almost 4 years. Things have been extremely difficult and rocky since having our baby. He does not help with ANYTHING. He runs and does what he wants, when he wants. We argue constantly. At this point, idk why I stay... I love him of course but he’s got too much growing up to do and won’t seem to do it. Our baby was not planned but she’s my whole world. Neither of us were ready, I know that, but I had to put my big girl pants on and grow up quick, as should he... but he won’t. Yesterday on my very first Mother’s Day, he left for a while to spend some time over at his moms with his family and came home completely wasted. Throwing up, passed out, all that fun stuff. ON MOTHERS DAY!? Who does that!?!? Like I’m at home taking care of our screaming, fussy teething baby and YOU get to come home drunk!?!? I’m at my absolute breaking point and have just not been happy for a while now... but I’m scared about a custody battle as I know his mom will be right on it if I leave.. how did you get through a breakup involving a baby/kids? What is some advice you could give me? I’m struggling 😥😔😣