Elgrand
If JSA is still available in your area then you can claim it and once your claim is accepted and put into payment then you can present a new Fit Note and be excused from any Claimant Commitment for up to 13 weeks.
If you are in a Universal Credit area then I would get face to face advice before make a claim for UC as it may not be in your interest to do so.
You can use the PIP reports as support for your ESA claim but even where there are areas of overlap the criteria can still be quite different so you need to explain the relevance rather than just sending the ESA people the report.
Gordon
Nothing on this board constitutes legal advice - always consult a professional about specific problems