Teesside have choice based lettings. Once you're registered, you're allocated a set number of points. You log on to Compass Housing every morning, bid on properties you are interested in and wait and see.
The person with the most points gets first refusal, then it filters down the list. I was number 5 on the list for my house but because there was a six month wait for the keys, the four before me refused it. I accepted the house within a week of going on the waiting list.
The more you bid, the more you are awarded.
There is no need for OP to present as homeless or get pregnant, there won't be any temporary accommodation. She can't flat share because 1) they are more expensive and 2) they are few and far between.
There is a house share in my town on the most prestigious street in town. For a modest sum of £130 per week, you can share a three story, ten bedroom, two bathroom house with nine other people. I pay £87 p/w for an entire house 
Can we try to remember there are whole towns and cities outside of London who do not share London's housing crisis?
OP, if you don't mind asking, how much do you earn? I'm just thinking you might be better off financially if you go into care work. I've just taken a job through an agency which pays £8 p/h for the first three months rising to £8.60 after the probationary period.