I really feel for you, I managed to get a move last year, only through a charity run housing association. I think you need to concentrate on getting your sons needs met, maybe him having a medical assessment would be beneficial to a housing claim, you may even get your banding upped.
I know where you are coming from with the lack of sleep! both of my sons sleep about 2 hours a night, since moving though it has gone up to 4-5 house which is great stuff, I can understand the stress of it especially with the lack of sleep, try to get a break once in a while by getting someone to watch your son whilst you do something for you (long bath or a sleep)