That's right about your stamp being paid if you sign on, which is one of the reasons why I have always signed on when I have been unemployed, even though the bastards at the job centre are so horrible to me.
I also sign on because I am entitled to 6 months' jobseekers allowance, so I am going to have as much of it as I can, until I get a job.
I recently sent away to find out my National Insurance contributions to date, because someone from here said that single mothers on benefits did not get this paid for them, and this would have meant that I did not have entitled to a full pension when I (finally) reach state pension age. But it showed that it was paid when I was on whatever the benefit was called, so, actually, I now have the necessary contributions to get a full state pension. That's good to know. I found out how to do that on the HMRC website, and it shows years when the full contributions were not paid, and how to make these up, too, if you can.