LL will usually agree where there's DA. They'd have to really really hate victims of DA not to agree.
Contact Women's Aid who can help advocate for you, report him to police and get a crime number if that's what it takes. Don't expect much of an outcome there but sometimes officials need the evidence that you're not just making it up to suit your own ends, so police reports are that evidence regardless of the outcome. I'm not suggesting for one second that there's going to be a trial or any justice at all. Don't stick around being abused in various forms just because you think you're trapped, you're not, speak up, be truthful about what's happening, don't downplay anything, ask for help, it is there but you have to engage with it. You can get out of this situation far sooner than you think.
Do you know, can either you or your STBX afford to pay the rent in full? If one can and one can't, the LL is going to be happiest financially giving the tenancy to the one who can. Or they may prefer to give the tenancy to you because you're not the arsehole tenant. If you want to move you can always move next year.
Be open to any possibility that means you don't have to live with him. Before you turn down any options, ask questions to ensure you understand the full consequences of that decision, for your situation going forward. Then decide once you have the full facts.
Have you been on Entitled To website? Enter your hypothetical situation into the calculator as if you were single.
There's a website where you can find out the LHA for your area, you'll need to do that first. Is it just you or any dependent DC? That affects the number of bedrooms you're officially allocated. Any extra bedrooms or any extra rent (for a nicer place/area) above the LHA then you'd pay for that part yourself, even if you got maximum LHA. Obviously you don't have to stay in current area you can move, but to find the LHA you need a postcode so maybe ask friends or colleagues if needed. Otherwise IDK if the postcode of a shop would work, you could Google a random address if it would.
Non-dependent DC would be homeless in their own right if at risk of abuse so could present at the council as such, or they'd remain housed with your ex if he's ok with that and homeless (and so eligible for help) if not. Non dependent DC who want to live with you may need to consider leaving education and getting a job to help pay rent/utilities/CT, if that's the only way you can make it work. Over 18 you're not obliged as a parent to house them so there's no help from officialdom to help you do so.
Depending on ages of DC you may be able to be housed in a refuge if you're the one to leave since you're officially homeless at that point, unless you can go out quickly and find a private rental (you sound skint so maybe not the preferred financial prospect for most LL) and from there into either private or social rented housing. I believe the rule for refuge is no boys over 10. That doesn't mean you're on your own and Women's Aid won't help you at all though, that's not the case.