HB is supposed to sort out your entitlement within 14 days unless there is a good reason why they can't.
If they can't they are supposed to make you an Interim Payment based on their best guess of what you are entitled to. The only problem would be that if eventually you were assessed as not entitled, you would have to pay it back, but that would be little by little if you didn't have the cash then.
My suggestion is that you go into the HB office, even if you have been there endless times before, and 1) ask for an interim payment and 2) ask if there is any information they are waiting for from you. Get a receipt for your visit which says that you made these requests and records the answer. So many times councils say they couldn't pay HB for such-and-such a period because the claimant did not supply information, when in fact the letter got lost, or they themselves mislaid or misunderstood what they had been told. Get absolutely everything in writing. Get proof of posting (it's free at the post office) whenever you send anything and get a receipt when you hand deliver anything.
As far as your landlords are concerned, stay in touch with them all the way. Even if HB is payable eventually, you will probably have to pay a certain amount, maybe to cover water rates or similar. You could ask your landlord for guidance about this, but start paying something now and pay regularly. I would suggest you start with £5 per week, and somehow it looks better to have £5 going in each and every week rather than £40 here and £20 there with nothing in between but long gaps, even if it adds up to the same.
So far as the money from your house is concerned, HB must ignore a certain capital sum - I think it's about £3,000 - then above that they take it into account but not pound for pound. CAB can tell you more. If you are thinking about your children's future, I wonder if there is any mileage in putting lump sums into trust for them when the house is sold? You would need some expert advice on that, but it is possible that it could be a way of keeping capital aside for them and not having it treated as yours.