Bloody hell, I'm exhausted just reading that!
This is my first pregnancy but I've suffered terribly with HG. As a result, I've mostly been in bed (have the luxury of that as no current children). I was signed off work and only recently easing myself back in working from home.
I couldn't imagine having HG, a child, work and being expected to do everything and have sex most days!
My DH works full time, has done all the housework, walked the dog every day and run around after me. He's gone above and beyond and I'm so grateful. I find your husband's lack of support shocking and sad.
And as for sex, we haven't even had sex since I got pregnant because of the HG! Almost three months now. My DH hasn't said a word because he knows how crappy I'm feeling.
It's been an exhausting few months, mostly mundane chores and sickness, but we're both just muddling through a rough patch, doing our best and knowing it won't be forever.
You need to speak to him and he needs a serious kick up the arse.