Sarah, the short notice can make it very challenging to line up a grant to cover your arrears. It may be possible, though there may also be some other opportunities for helping you maintain safe housing for you and your two boys.
Due to the complexities of your case, I believe that a meeting with a social worker is the best starting point. They can help you with housing stability, since that is most important for family stability.
It will be important to know that you will be able to afford the rent on this home or any other should you move to a new place. If rent exceeds 40% of your monthly income, then it makes it incredibly challenging to pay this on time plus all of your other living costs. I realize that children are not cheap, so it really takes discipline to keep living costs as low as possible.
There are also these Durham aid agencies that can match you with a case manager. If a grant is possible they can work quickly to find a way to make it happen. Please keep your mind open to other options, since those could be more sustainable in the long run.