You need to contact your local council. They will have a points system which they use to allocate priorities within the council housing system. As someone that has basically been made homeless, you would come fairly high on their points system. All the same, you may have to wait a while to get council accommodation, so have contingency plans in the meantime. Can you stay with a friend or relative? Are there any hostels nearby? Maybe you can ask your local council about housing options for you while you are waiting for them to house you – it is their job to help you and they will deal with a lot of queries similar to yours – they will surely have some helpful suggestions.
Re. your financial situation – I’m pretty sure you’re right that you can’t claim JSA while you’re a full-time student. All the same, ask the Job Centre if there’s any other financial help they can provide you with – for example, they might be able to offer you a small Crisis Loan. In addition, keep trying to apply for part-time jobs… something like catering work might be good, if you could do it in the evenings after sixth form.
This is a very difficult situation you are in for someone so young and I feel for you.