HR Specialist here with several years of recruiting experience. Let me try to give you some advise.
Never evoke the feeling of desparation. Never beg. Never propose working for free or something like that. Never make the recruiter feel guilty for not picking you. The recruiter will set up an interview if the job description matches your skills. It doesn't matter how desperate you are, you must never let that show. The recruiter will want to hire you because you fit the job, not because of pity.
You already realised yourself that finding a job is a matter of timing. So never take any declination personal - it just wasn't the right time. Never give up, just keep on checking career pages and keep applying to those jobs that you have exceled in.
As a matter of fact, it is definitely true that you are in a downward spiral if you are long term unemployed. This is not an illusion, this is fact. However (again), this is not your fault, so you don't have to blame yourself. Just keep applying. It will work out one day.
The recruiter will come to the conclusion that no one else gave you work before so surely something must be wrong with you. This is a very prominent bias and is very hard to not fall to. Many people fall to it, especially if they don't know about it. Whenever we see something strange about a person we don't know, we always tend to assume the easiest explanation possible, which is in your case that something must be wrong with you and therefore nobody wants to work with you. It's a simple prejudice building process that many people can't shake off. However, this is not your fault, you must be stronger than that.
Easier said than done, right? Still, keep on applying and be confident. It will work out. Turn that desparation into desire. You might have no major experience but you are very eager to learn, you will work hard and you will want to prove yourself worthy. These are things that you can mention in your introduction letter.
By the way, the letter is overrated anyway - what matters most are your skills. Highlight what you've got and what you can do. That's the interesting part for a recruiter. And always stay confident and never give up. I believe in you.
