ok, easier said than done, but ignore the negative comments
I, personally, don't know anything about benefits. I can't answer any of your questions about those. It would in any case be better for you to get bespoke advice, which is why you are being advised to make some phone calls and get some proper advice.
However. Although I don't know anything about benefits, I am a bit confused by what seems like a mismatch between your insistence that you are too ill to work more than you already do, and your children are too ill for you to work more than you do, and that this is formally recognised in a variety of ways - and that there is no point in applying for benefits on this basis.
You say there is no point as it is so hard to get awarded it. Well as I understand it, the system is supposed to be you either work, or you get benefits (or you are married and your spouse agrees to support you - let's leave partners out of it though as yours is part of the problem, not part of the solution). I know the system isn't perfect but you need to engage to have a chance of getting anything out of it.
Start making phone calls, start following the leads we're suggesting. Not because they are perfect or easy or because anyone here knows everything about the perfect person to help you, but because you have to do something.
What can we help with? You need to pick up the phone. how can we help it feel less insurmountable to do that? (or if not that, why aren't you doing it?)