Partner issues- anyone else in the same boat?

Hi all, Just wanted to reach out and vent for some support really. I love my partner to pieces and we are recently engaged. We have an 11 month old son together and it’s been a whirlwind of a year… However, my partner often doesn’t give me a break from my son and I’ve been out of the house alone 3 times in 11 months, mostly for walks and one night out for a meal with a friend. I feel like I’m the default parent keeping lists in my head of appointments, development milestones, nutrition, activity ideas, childcare, funding. I cook from scratch every night and I try and keep up with the cleaning. I try and chat with my partner about that I need some time to myself and he says to let him know when to book something in around his schedule. His schedule changes weekly and I barely got enough time to arrange anything so weeks go by and the same cycle repeats. When I ask him to do something he says he doesn’t know how or what do I want to do about this or make a list for him. I’m burnt out. When I try and communicate he shuts me down or goes on his phone. His mum and sister have been on my back about letting him lead more and that I’m controlling. However that is so far from the case, he barely picks up anything after himself. He still has a life, full time job, goes out to gigs with friends, goes for a run/swim/gym etc, and I’m sitting at home with a teething child with hair hasn’t been washed in 3 days. When I try and bring up what his family is like to me making comments he says that it’s all in my head and that they didn’t mean it that way. E.g. MIL saying I need a breast reduction because my boobs are unsightly, SIL saying that I can’t go upstairs to hide eggs for the Easter egg hunt because her kids underwear is left out, like wtf??? I feel like I’ve invested in so much into this relationship and I’m tired. I don’t feel valued or appreciated. Anyone else in the same boat as me?