Why do people keep saying private rent? The OP is trying to private rent, through the housing officer facilitating that through the PR scheme, where they give you a deposit. The problem is there's a severe rental shortage at the moment, and so landlords are not so inclined to take people needing housing benefit, even thought the law does not allow them to discriminate, plus with 30 people going for each flat, the chances of being successful are low.
Private rental is incredibly stretched in most places right now, and the OP and her partner may not be even standing a chance in London, by the time their housing officer finds somewhere, it will have gone.
In the past councils used to run schemes to help landlords rent to tenants on welfare, as well as provide minimum standards for the tenants, like all their gas/electricity checks. Our council does not do these any more. The OP and her partner will have to take their chances on the open market, and even with a housing officer, may not get anything.
OP, you should not have turned down the temp accommodation but you know that now. You could have taken it, and then still sofa surfed for most of the time at your friends.
I don't like the tone of many of these posts, Labour hasn't promised to build over a million homes for fun, and moving is no guarantee of an immediate job, and if there's an issue with one month's rent, they may not find it that easy to move and start up elsewhere.
Do contact Shelter, OP as the law is quite complex around this plus many of the officials involved don't know the law either and will tell you all kinds of things to try to manage the demand and make you go away. You need to know your own rights, and also what is not possible for you now.