You sound like you're having a really hard time, Hedgehog, & I think there have been some quite nasty comments. It's just the phrasing about whether to "let" your husband go out that doesn't sound too good. And if you're admitting to irrational jealousy, when your dh has given you no reason for it, then obviously that's something you need to address.
Having said all that, you've got four children with health issues, you're still breastfeeding, & you should be your dh's top priority. When our kids were ill (& we've only got two, & they don't have ongoing health issues) neither dh nor I would have just assumed it was OK to leave the other person to deal with it all. I can't get my head round the idea that your dh left you with all this to handle by yourself to go on a 10 day holiday in the Summer - I think that's actually way more unreasonable than a single night out. They're his children, too. Hasn't he ever been supportive / done his share? I'm not surprised you feel resentful & unable to cope.
In this particular instance, is some compromise possible? Does he have to stay out overnight? And why does he have to get drunk? I don't actually think it's very responsible for grown adults with children to go out & get hopelessly drunk, anyway. Have you told him how you feel - because, again, I can't understand how a loving partner, on being told that his OH was at the end of her tether & unable to cope, would just bugger off for a night out.
In the long term, you're going to need some more support, OP. It sounds like you've got too much on your plate. And your DH needs to pull his weight.