I don't think you're being given correct advice by the Council.
If you have nowhere you can go, they should re-house you. If you have a house but are unable to live in it, through no fault of your own you should be seen as unintentionally homeless. Because you have kids, you should be seen as vunerable and so you should be housed. The thing is it will probably be in a temporary hostel which may be completley unsuitable for kids. EVENUALLY you should get a council house.
Make an appointment with the Homelessness section of your local council. Also talk to your CAB as they will be familiar with your local situation. Alternatlively many councils run a "housing help" centre.
You should also be able to get housing benefit in a situation like this, epecially if you are in an abusive relationship. Other alternatives might be trying to get on other social landlords waiting lists. Many councils run a bond scheme (bascially they give you your deposit) which might help. You might not qualify however as they often exclude people who should be housed by Homelessness.
As I recall, the fact that your relationship has ended should be enough to justify your leaving the house, let alone the abuse.
Another alternative is to get an injunction, change the locks etc. Presumably he SOMETIMES leaves the house?
Mind run an organisation called Women in Mind. They are geared up to support women-eg all the project workers are women,. They seem to work closely with Womens Aid, NSPCC (for support for your children) etc. My experience of them is that they are fantastic, they really offer people a lot of support and people can't say enough good things about them. In our area there is a waiting list so an alternative would be Women's Aid, who also offer support workers.
I really hope it gets better.