Bloody hell.
YANBU at OP. By contrast, your DH is being VU & a complete shitehawk.
What's this business about him being unable to get up with your DC during the night? My BIL doesn't get up with my v severely disabled niece during the night but that's because he is out at work (& really is at work). His nights off he will share any get-ups, because he is an engaged & loving parent who doesn't think my sister being primary carer absolves him of any/all responsibility for his children. I do understand it not being ideal to have a broken night followed by a full day at work, but even if you're not working, if you're providing high-intensity care for your DC you can't afford to be doing so 24/7 because you will burn out.
This business about you not spending any time with your friends once you are able to go out but instead only socialising with him is a HUGE red flag. Bobby could have stopped the train with it in the "Railway Children" with ease. (And without the loss of her petticoat, you'd hope the train company discreetly replaced it for her... but I digress.) Anyway, that kind of controlling behaviour is totally TOTALLY unacceptable (not that he's covering himself in glory otherwise) - is he attempting/has he attempted to interfere in your friendships in any other ways?
He does seem really set on totally controlling your relationship: he will do as he pleases; he will lie [by omission]; when caught in his lies he will blame you for them; & it sounds like he's painting a VERY different picture of things to everyone else.
Serious as him fecking off out when he told you he was at work is, it seems to be one thread in a knotty tangled mess of issues that you need to look at. With, in all honesty, a view to whether or not you feel your relationship is tenable/salvageable.
Please don't let him mess with your head. I certainly don't want to sound like I'm advocating for you to simply sling him out/you to walk out on him. But his behaviour towards you goes beyond simply being disrespectful - it is manipulative, deceitful & controlling; & if PPs are right re an affair/sex with prostitutes... does he have any good qualities?
Do you have any RL support close by you can talk to about all this? And, crucially, enlist to help in practical ways if necessary?
Please be gentle with yourself 