Contribution-based JSA is available to anyone who:
(a) is genuinely unemployed ;
(b) is genuinely seeking work ; and
(c) has been paying National Insurance contributions.
It's not means-tested. (Even someone made redundant from a million pounds per year job, with several mansions, a yacht and a private jet is entitled to it). The maximum period that it can be paid is 6 months.
Income-based JSA can be applied for by anyone who meets conditions (a) & (b) above but who is not eligible for Contribution-based JSA (either because they've not paid enough National Insurance or because the 6-month period for Contribution-based JSA has ended). It IS means-tested, with your savings the income of your partner (where relevant) being taken into account.
So you should be getting Contribution-based JSA but you need to be aware that if you haven't found a job within 6 months you might not get Income-based JSA thereafter (depending upon whether you've got savings and/or a partner with their own income).
If you don't pay rent you won't qualify for Local Housing Allowance payments. If you've got a mortgage you can't get Support for Mortgage Interest until you switch from Contribution-based JSA to Income-based JSA after 6 months (if you're eligible for I-B JSA).
Benefits such as free NHS prescriptions only apply to those on I-B JSA, not to people in receipt of C-B JSA.
The only other financial help you might be able to receive is a reduction in your Council Tax. That's means-tested and, as it's no longer a national scheme, the reduction you get varies from one local authority to another. If you live in an area covered by both district and county councils (rather than a Metropolitan or Unitary council), it's the district council that you need to apply to.