Yeah, I'm afraid you can't be honest at all with these morons if you need the money.
Just toe the line - apply for all the shitty jobs millions of miles away, keep your proof, make your cv shit, turn up to the pointless courses (and get sent home because you're over qualified), answer their dumb (repetitive) questions.
Don't tell them about the GOOD well paid jobs you're applying for in your field - keep feeding them the same old shit - until you get your start date of your (great) new job - and don't tell them the start date until about 2 or 3 days before because they'll keep telling you to start trying to find a shit job with a sooner start date.
Keep a shit cv for the shit (usually fake) jobs, and concentrate all your time and effort on the real job hunt.
In all honesty the whole JSA is a complete joke for people who might need to pay rent and bills in between jobs (I was made redundant in the recession, along with about 150 other employees in our branch).
I had a job lined up within 2 weeks, and they still gave me shit about needing the help.