Dad isn't interested in caring for baby/relationship problems

While I'm looking for some opinions on what I should do or think, I just want to rant at this point. Long post.

Me and my fiancée got together 4 years ago and we are quite young(I'm 20 he's 22), we moved in together 2 years ago and had a baby girl 4 months ago. I usually am quite independent and would do cleaning, cooking etc without asking for help. He's always been a good, caring man etc. During my pregnancy he was so happy, the baby was planned. But recently I feel like I'm completely on my own with everything.

I do everything around the baby by myself, I give her baths, feed her, make her bottles, change her, put her to sleep etc literally everything. I clean the whole house myself when he's at work, wash his fucking socks, iron his shirts etc. I feed walk brush and bathe the dog by myself too, all while looking after the baby day and night. Now I understand if he's been working all day, when he comes home I want 10 minutes to myself, whether it's to go pee, or make and drink Tea, but I have to actually beg him to take the baby for a few minutes,as he won't usually even bother to take her himself. When she needs changed, he ignores the dirty nappy Nd passes her onto me even when I'm doing something. When the dog is barking because she needs outside, he won't even bother letting her go outside. When I ask him to do something small like picking up something from the floor he gets annoyed and saying hell do it soon and never does it and it's only been like this recetly. The main problem is surely xbox, as soon as he wakes up, hell say a blunt hello to us and switch the xbox on, won't even grab his daughter for 10 minutes. When I have calming music playing on the TV while she's falling asleep, hell turn it off and start a game. When the baby is starting to get fussy, he'll keep playing until she starts screaming. We came in from a walk last night Nd I asked him to take her hat Nd coat off, as I had to go to the toilet, I came back 10 mins later and she was in the pram still in the hat etc and he was on the xbox not seeing the problem "because she isn't crying".

Anytime I bring anything up or start talking about him not helping me, he starts to blame it on me saying I'm just being a dickhead or starts yelling asking me what my problem is etc.

Today I got mad because while he's not even been liking photos of his daughter that I post on Facebook at times, he went out of his way to like a photo of a girl from September. He says he didn't even mean it or remembers that he liked it etc, and automatically started yelling etc that I'm jealous and paranoid(all I did was bring up the fact that I seen it and asked him about why he purposely liked a photo from months ago of a girl he supposedly never seen in person but she works in the shopping centre where he works too and he played dumb) then instead of talking like an adult, he cleared off to go somewhere for a few hours, he says he was at his mums house and took a taxi down(last week he had a problem with me getting a taxi to a hospital for my baby's appointment while it was raining yet he spent 40 pounds on a taxi to his mum). I recently got diagnosed with random fits of some sort so it's not ideal for me to be alone with the baby before I start medication because I could be holding her and start shaking and faint etc, yet he left me alone while I'm drained and cleared off. He then came back not talking to me, and when he started talking to me he was slurring his words and I just knew something was off, he then changed his story and said he doesn't understand what the problem would be if he did go out of his way to like a photo. BTW I don't mind him talking to or liking other girls pic, and he knows that, so idk why I'm being calling paranoid.

I then asked him to take out baby so I can make her milk, and he got annoyed because he wanted to go to bed early.

Even last week I had a hair appt and was supposed to get my nails done so he was home with the baby. Or so I thought. I came home early as the nails took less time, just to find his mother home alone with our baby without my knowledge. When I asked her where he is, she told me he went to get a haircut too and to get new clothes, fair enough but I trusted him with staying home with the baby and I knew nothing about him going anywhere.

I really do love him, but I don't even know whether I'm happy anymore, and I do completely everything on my own. When I clean the house, he comes home and won't even put his socks in the bathroom or tidy after making food, and then it's me who has to do it the next morning. I don't have an option of moving out or anything so idk what the fuck to do. Yes I love him, but I dont want to be on my own for the rest of my life.

Before I even got pregnant, we talked about finances and bills, since he earns a lot for his age, he is the one who said he will pay bills, and send me about 200 pounds a month just for my expenses and if I need to go get food for the house or milk for the baby. Everything that is in the house was bought by ME. Everything for the child I bought or else was gifted from family. He doesn't mind spending 50 on a game, but will get fussy with spending 10 on a new blanket or toys for the baby. He knew once I'm pregnant and have the baby I won't have any income and it was his idea that he will do all that. Here I currently am with about 50 pounds in my bank to do me until xmas, that I earned by selling my clothes that I don't wear, and if I need say 4.50 to get dog food, he will literally send me 4.50. I am dependent on him but can't do nothing about it at all. His mother isn't fit enough to look after the baby if I was to find work. Yet he comes home and asks me why there isn't any food made for dinner, if I don't have say chicken I can't even go get it?

I'm just miserable, but want to somehow fix all this. I know I could be more happy on my own, but I can't do this right now, unless someone has any ideas? The worst thing is, we argue, and after a day he thinks everything is fantastic and doesn't see any problems.