You would likely only get a room in a shared house following a stay in a run down B&B/hostel. Been there when illness/erm, gross misjudgement led to me being on the bones of my arse, pregnant, living in my van and in a very shitty, vulnerable situation. I agree with PP saying that a baby and shitty circumstances are the golden ticket (depending on where you live) although I am unsure whether this was said in a sneery way. It is not, however a golden ticket to a golden life; shitty neighbours (and I mean awful shitty neighbours), skintness, total loss of self esteem, being treated like crap by the council are par for the course in my experience along with having to prove and re prove you are ill/in need/homeless relentless form filling/question answering etc etc.
Will you present at the council offices as sleeping on the street as that is the only way I can see this working for you, then taking what is offered (emergency accommodation is usually dire and will no doubt be what you get for the foreseeable) until you are 35 and eligible for a flat/bedsit, assuming g the rules haven't changed by then?
It cost me a fortune to do my council house up to a basic standard (which I got by way of a B&B then flat). and, not having a fortune to spend, has taken much saving, going without, scrimping etc etc to provide my daughter and I with a washing machine, tumble drier, fridge freezer, cooker, television, beds and bedding, towels, kitchenware, flooring (£40 a room to have even the crappest cheapest flooring fitted - I was lucky enough to have a friend who fitted it for free - total bodge job but good enough - had to pay out for the stairs though as they have to be safe! not to mention carpet glue, those spiky carpet thingies, laminate for kitchen etc etc.
3 years later, my house is starting to look halfway decent. The multiple tins of magnolia I initially used (and have since just painted over as needed and certainly weren't 'free!' I think it was £20 a tin of no frills Dulux), paint brushes, rollers and trays, curtains, furniture from charity shops have added up to a hell of a lot more than you'd imagine, not to mention bills and ongoing expenses (landline as mobile phones are unreliable here signal wise and I have DD and ongoing health problems so need to be able to contact/be contacted by various folks).
I receive ESA support and am basically fucked and getting progressively worse. I am also in receipt of PIP, Housing Benefits etc and live in constant fear of the next assessment/hoop to be jumped through and the imminent change over to Universal Credit, which actually makes me feel suicidal. I used to have a very well paying job and was blatantly misdiagnosed by 4 Doctors at my old surgery and sent away when I couldn't speak, was incoherent and could barely walk (brain illness). The hospital then took 5 attempts to do my spinal which, it is reckoned has caused wide spread nerve damage, along with problems following my illness, depression, anxiety, and am classed as disabled. I am boring on about this to illustrate that, although I am very ill, which is well documented by professionals/appointments etc I am still living in constant fear that the rug will be pulled from under DD and I by the ever shifting goalposts. As such, in the ESA work group, I don't know how you can even consider that you are in a stable enough position financially to move out and I say that kindly. You could be declared fit to work at any time then really be up shit street if having to pay all your bills, possibly having used credit to sort a place out. Get back into the world slowly. You are totally different to your friends. I fretted that I did the whole thing back to front, marriage, split up, left stable home work etc (albeit in immediate aftermath of illness) and am the token single parent in our circle of home owning, married parents, however I've come to realise I have a massive amount of freedom my friends don't, financial, physical, and I am sure by the time the children are 10 or so there will be at least a couple of divorces whereby friends will find themselves in the crap situation while DD and I are hopefully still plodding along happily. I kind of see life as a snakes and ladders game, all unexpected snakes then the odd ladder if you are lucky! What i'm trying to say is don't judge. Take life at your own pace and don't compare. Get on the housing list because, why the hell not?! Get on some medication which maybe helps you feel more able to get out and about, volunteer again etc, small steps, but be kind on yourself. Illness may hit someone else at any time so please try not to feel life is shit; you clearly want to better yourself so goo for you (what about an online course/education?). Sorry for the utter garbled essay but I really feel for you and to anyone who wants to benefit bash, I could quite easily have sued the NHS for all their fuck ups which have essentially ruined my life but would rather eat my own liver than do so, so make no apology that I am on benefits through no fault of my own whatsoever (I miss work so much it makes me cry).