Stuie:
It sounds like your local Jobcentre 'adviser' is making it up about what counts as looking for work. I too am on JSA (contribution-based, when that runs out I won't qualify for Income-based JSA - who knew there were 2 sorts?). Anyhow, searching job websites, signing up to recruitment agencies (which as I'm sure you have found out can be quite a long winded affair in many cases), are all accepted by my Jobcentre as activities in the 'looking for work' arena. Obviously these are the PC equivalent of 'looking in the paper'.
But I don't find it too hard to send applications (CV and a bit of spiel) in response to at least 3 jobs a week, but they are all over the country. So what? I know I won't get much interest, if any, because I'm slightly out of the criteria (too many employers being over-fussy about 'relevant experience, but that's another rant), and I probably have no intention of actually moving to another part of the country for any old job, but it keeps the 3 things a week tally up. Also, some jobs are advertised in more than one site or agency, they count as no-one is going to bother to check, and there is no actual proof they are the same job - easy hit. The form they give you to write in is pathetic - I started out keeping a thorough and useful job search log in my own format, electronic of course, and printed it out. No, not good enough even though it was 10 times more informative. So I now scribble in the boxes as small and untidy as I can, and I can see the person only glances, doesn't actually READ it. I also try to get lots more than 3 crammed in, as they are even less interested in reading it then. I do have the actual records should I be asked for them.
I gather they get fussier after the first six-months, and start expecting you to apply for any old job, moving away from your 'preferred' role. I won't be crossing that bridge.
As for their help with CV writing - I was chastised for not having my 2 English GCE O levels (1967!!) on my very decent CV, which has been perused constructively by many and got me many interviews. When I said all advice is that such old and irrelevant information is neither necessary nor recommended, this was tut-tutted. This has not given me any confidence in their competency. As said, they are not interested in helping you get work, only about their system and targets. A friend who is 6 weeks from her state pension age, has been sent on a 3 week "job improver" type course, they know my friend will attend but others wouldn't, and then she will sign off (being retired), thus target met, and tax money wasted.