I have been in the same situation. Pressure to have sex (that I did not want, no feelings that way left) to keep the marriage going as HE needed sex. And it was all my fault.
Back story, he raped me after birth of first child. I suggested we tried it, it hurt, he ignored me when I told him to stop.
Also had youngest who didn't sleep at all for 2.5years, I to was exhausted.
He was also (as your dh is) emotionally abusive.
He did absolutely NOTHING round the house, inc bills, arranging repairs, diy (he oiled a door twice in 8 years...woopeedo).
As above I was exhausted and after the rape I actually didn't really want him near me. I did not trust him. He also had sex in a porn star way...I look back now and its like he mentally had an audience...!
I too had duty sex, made me feel like a prostitute too. I only did it to keep the marriage going for the children, I didn't want to be a one parent family.
He also did not allow me to get out and about, I had one night a fortnight (reluctantly given) to go out, but if he decided he wanted that night, he would just tell me at he last minute 'no i'm going out its x's bday/xmas night out' etc and go, leaving me with children again.
He never did anything special to make me feel special/wanted/loved/cared for, no hugs, ever.
He kept all our savings in his name, so I had to ask for extra money to get repairs done etc, apparently I should have saved up out of my housekeeping money. It was also not enough for clothes for myself or shoes, just had to go overdrawn. And he earned A LOT of money.
Oh and he never ever praised my parenting of the children.
He made me feel like it was all my fault that I did not want sex.
It was a sexually/financially/emotionally abusive relationship.
When we went to separate he blackmailed me to go to relate about our relationship, he told me that I dare not go as I was too embarressed by my behaviour (lack of housework/poor parenting) and she would expose me. Bloody didn't, it exposed him, end of the first session she had me speaking to womans aid and the HV.
He told me that the reasons for getting a divorce from a such a good marriage were so frivolous that most likely I would not get any benefits.
He emphasised how hard it is to be a lone parent, over and over again.
He's a dickhead.
I got lots of support from relate, hv, social services, domestic abuse servies (tho they do back off after threat is over), gp's, police, friends (always always speak to friends how you are feeling DON'T shut down! It weakens your resolve and you end up listening to HIS voice instead of theirs), the local sure start centre. They all helped me.
It is not your fault for not wanting sex. He should have been a good enough husband to you. But he wasn't. He was/is an abusive husband.
Use a free half hour to speak to a few solicitors (that way you also get an hour and half of advice, most of it the same, some different and helpful ;) ). Go online and use the benefits calculator. Housing wise, speak to your housing association, explain your circumstances.
Whatever you do, get out. Listen to your gut feeling. This is not right what he is doing to you. Get angry. Get really angry with him. And use that anger to fight back. You CAN do it.
As for the children, I found mine were better after he went, not immediately better, but we are a close unit now, he used to divide and conquer with us. He tries that still, but as I have them the most of the time...it doesn't work quite as well.
Oh and sex wise, its been 2 years ish, I still joke to people that I don't need contraceptives anymore, I just remember my ex. He's put me off sex for life.
Anyway. C'mon, think all this through, LISTEN to what makes sense from what people have said on here. And when you are ready, start googling. Don't listen to his voice, its hard I know when they are digging at your deepest fears, listen to ours, we'll help get you back up again and get your happiness back :).
And have on repeat