you will be seen as intentionally homeless if you don't get the LL to evict you, if you just pack your bags and leave then this often seen by local authority as making yourself homeless.
I would go to local council and see what their process is, they will most probably want you to go on the council home/housing bidding list and bid for a home
how many bedrooms you need will make a difference to how quickly you'd be housed - 2 is often the sweet spot
one beds are the most difficult followed by 4 beds
Also take the paperwork with you, it's surprising how often LL don't do the paper work correctly and then when it gets to court they have to start again - right back at the very begining. Having council look at the paperwork will give you a clue to whether this might happen.
for a while I worked in council office and often people would come in complaining that they hadn't been offered a home, the main reason was, they hadn't been bidding and consequently if they don't bid they wouldn't get a home. Make Sure you know exactly what you have to do and make sure you bid every week if that is the system - its often done on an algorithm, so the more you bid the higher up the list you within your band.
I expect the LL doesn't want the deposit to be returned - have you checked the LL put the deposit in a proper scheme - if its not been done then they can't evict you. Do you have a gas safety certificate if you have gas? again without this you can't be evicted