OP I think you're in a catch-22 situation at the moment - it's a toss up between a job vs temporary housing, and obviously getting a job can help you get secure housing but your insecure housing is getting in the way of securing your job longterm.
I think your situation has highlighted how unhelpful the UK housing/employment system is - logical solutions (e.g. living in a caravan on someone's driveway) have essentially been outlawed (you have to apply for planning permission to do that), which makes it hard to claw yourself out of your situation. And then making secure residence a condition of employment means lots of people just get trapped in a situation where they're both unemployed and homeless.
In your position, I'd probably be tempted to give the job a go AS LONG AS IT'S NOT A VERY TIGHT-KNIT INDUSTRY - if this job doesn't go well, then it could ruin your professional reputation forever, which wouldn't be worth the gamble. But if it's something you could potentially get away with for a month or two whilst saving and hunting for a more secure/tenable housing solution then I would also be tempted to go for it.
I think you have no choice but to reach out to places of worship/community centres/local bulletins etc to try and get some accommodation in place, even if it is a small town I think most people would be sympathetic to your situation rather than judgmental as long as your phrase it correctly - e.g. when I struggled with getting a rental where I now live (I was relocating from abroad), I posted a bio on Facebook asking for a room and was offered one by a landlord who'd not yet put their flat up for rent yet - not all flats will be advertised, and there are kind people out there who won't want to see someone struggle when they have an asset available. Make sure your bio only covers factors highlighting your stability, only mention casually that your short term tenancy has come to an end. Don't mention the CCJ preemptively etc.
Best of luck OP, your story really makes me despair for the state of this country.