OP, my DH was a bit like this to begin with. What helped was, even from when our DC was a small breastfed baby, walking out and leaving them to it. After I snapped one morning when DC was around 6 weeks after doing it practically alone up till then (DH was working long hours at work and getting back at 2/3am, every night), I just walked out on Saturday morning, texted "bottle of expressed milk in the fridge" and went out for breakfast and read my book. DH managed to survive the morning with baby DC, surprise, surprise!, and brought him along to the cafe I was in when DC started mithering for a feed and, after that, it became a regular thing. DH would have DC on Saturday mornings and we'd meet for lunch afterwards and I'd feed DC. He also stepped up at other times as he realised I was at the end of my tether. From DC being around a year old, we'd split the weekend - one person would have a long-lie then take DC to the park. The other parent had an afternoon nap. And the second weekend day was family time. It's not perfect (DH is always at work during the week so I do everything then) but it's enough to stop my resentment building to levels where we have no future together.
And every couple of months, I get a child-free weekend, either because DH takes DC to visit his family by himself or I arrange to go away with friends.
In our case, we were both incredibly stressed. In your case, your H sounds lazy. But my experience is that you can't wait around hoping that your OH will have an epiphany and realise they need to pull their weight. Instead, you have to shout your needs from the rooftops, draw your boundaries clearly and be so unpleasant and insistent that compromise becomes the preferable solution for your OH because there's no rest to be had anymore in shirking their share.
And it gets better... the more they do, the more they take responsibility for, instead of assuming it's all down to you.
And if they don't change, well you've tried (not that you should have to) and it becomes clearer that instead of marrying an equal with whom you can share a good fulfilling life, you've unfortunately ended up with a sulky, grumpy man-child who was looking for a substitute "mummy" rather than a life partner.