Really sorry to hear about your loss, OP.
You need to put yourself and your DCs first - I hope you can work something out for the sake of your step children, but I do not think their step mother should be calling you names and trying to emotionally blackmail you at this very difficult time. Is there anyone more neutral who can speak to her and get her to be more reasonable?
Re benefits, you will be entitled to Widowed Parents Allowance and a funeral payment. You can apply for this through the Department of Work and Pensions. Please see link below:
www.gov.uk/browse/benefits/bereavement
I am a widow too, but five years down the line. You will be okay OP, and so will your children and your step children. I recommend the Merry Widow website for information and support:
www.merrywidow.me.uk/
I would also suggest joining Widowed And Young for support:
www.wayfoundation.org.uk/Screen/Home.aspx
They have branches all over the country and meet up and support eachother.
For support for your children and step-children:
www.winstonswish.org.uk/
For yourself, go to your GP and ask for bereavement counselling, or try Cruse, www.cruse.org.uk/ who provide support groups and one to one counselling, they will come to your house.
It is not correct that anyone who your DH was supporting at the time of his death has a claim on his estate. As his wife you are the main beneficiary. If you are concerned about it, consult a solicitor later on. For now, take very good care, and get all the help and support you can.