@massivestress well done for managing to keep largely calm in such stressful circumstances.
When you describe the handwashing, like someone upthread, I wonder if you are charging rent/from his PIP/or universal credit. I have been charging my son for his keep for the last 6 months, though he isn't on benefits and it is coming from his savings. So in many ways I feel embarrassed to be charging, though it is only going to actual expenses I have more of now he is here. BUT the big difference it does make for me, is that when he is being wildly annoying, I can remind myself there is a contribution to divert my frustration, or to get a small treat for myself, or a coffee in a nice cafe to give me some respite, or a treat for us both, like a takeaway, so we have a little bit of uplift, alongside the snarking. And it is also a tiny bit of a boundary, to remind him he is no longer 14, and that as an adult, there are shared costs which it is up to him to contribute to, if he isn't absolutely destitute. It is a very very hard situation you are any, and anything which helps releave the pressure, like counselling for yourself, or perfume, or a fun night out, or bedding plants or a reiki session or whatever, which his keep might contribute to, could help.
When they describe themselves as having no hope that any of this will change, of course it massively kicks you in the stomach, both as a parent, and as the person supporting him. But you know he is depressed. His judgement is skewed on this as on so many things right now, that doesn't mean it is the truth or how things will have to be. In my life I do find it helpful to look back on previous times of turmoil and to see how I am in a different situation now, against hope. And I do think tales like this can be helpful to our young adults, as they have such limited contact with the outside world, and so knowing that eg 13 years ago, I was struggling to get a divorce, to find the cash to pay for it, to get my ex to agree to it, to parent two deeply traumatised teens who weren't settled at school and knew no one. And over the next few years, xyz changed (even if it is no longer the case) and things did get considerably better. E.g. for my son who arrived knowing no one, first he went to the playground alone, then he met one person, then he dared to hang out with them at school, then over the next six months xyz... Of course none of this is now but it might help remind him of when he had agency in the past, and how small courageous actions then made a difference. Either way they might be small stories he returns to in the dead of the night.
Oh @Dogaredabomb I'm so sorry. This sounds such a stress. Irrespective of how angry your oldest son is at you sharing the situation, this doesn't mean you can't or couldn't. We are allowed to share what is going on for us, we don't need permission. I'm guessing he is so angry because he feels ashamed or scared, - not that this changes things for you, but you are allowed to be your own person, not just a caregiver in all this.
And similarly, with your youngest, so sorry he is now so scared. Does he have other support? Is there someone else he feels safe with who might reassure him or give a bit of respite, so it isn't all up to you? But again, we are all fallible, we make mistakes, it is really hard you have to manage the repercussions of this, but we'd all on this thread have done similar. You can't be perfect all the time, cut yourself some slack. Here are some virtual MN brownies for you to eat, all on your own, no sharing allowed.
In case it might help anyone, I've been reading Failure to Launch, Mark McConville, which was recommended on a fb thread I think. He is a clinical psychologist and family therapist with teens/young adults/parents. I'm around halfway through it and have found his tone, the examples and ways for framing things useful. It isn't necessarily dealing with the most extreme situations which have been discussed on this thread, but nonetheless, for me was helpful in surfacing issues in my control I could address.