I'm guessing the resigning is because they know where you work and you don't want them finding you?
My advice would be to do this if you feel it will make your life better - but to not do this in a knee-jerk reaction way. Plan it well.
If you need to leave your job as well, then I would suggest as a first point of planning to find a new job. Maybe in a new area, even on the other side of the country if you need to - but find a source of income to base your new life on.
Once you have secured your income, then find a place to stay. Having income will allow you more choice. Having no income would mean being landed wherever your authorities/welfare agencies put you, and many people find such places worse than where they started out. I have volunteered at hostels and homeless shelters, and it isn't just people in financial or family hardship that end up there. They also house (at in my experience around 20-30% occupancy normally) people who are on bail and not allowed to return to their family home due to their own violence/drug taking/abuse etc.
Please find yourself an alternative before walking out. I would hate for you to have to stay in such a place.
It might be hard to deal with the delay, but I can't imagine that finding yourself with no income and no safe, secure housing would leave you in a position from which to start the rest of your happier, successful life.