The only way to solve this is with a mind map and some serious thinking.
Two circles, 'stay' and 'move', and then your thoughts on each, radiating out. From that, you can move onto 2 columns, 'stay' and 'move', with arguments for and against listed.
Basically, your big issue is 'secure accommodation' versus location. Which matters more to you?
Secure accommodation probably means moving. It might not but it probably does. It also comes with a very unsettling degree of loss of choice and autonomy. You don't have much control over your secure accommodation (it's not that secure).
In order to get secure accommodation, you have to be prepared to give up a job where you are comfortable (OK people to work with, OK school to work in) and hope you can later find similar.
You have to give up some control about choosing where you live (to the extent that you may not only not be able to choose your area, you may have to live in a completely different part of the country - losing networks of support).
So there is a big issue about loss of autonomy and lack of choice. Along with losing job and networks.
Those are big things to lose, frankly.
On the positive side, secure accommodation is great, if you can get it.
But ... in this situation, is it secure? It's not secure if you find that the factors that you gave up in order to secure it can't be replicated (job you're happy in; support networks;feeling at home in your location).
They might be - but you have no certainty.
Private renting can be hellish and expensive but it does have the advantage of returning some choices to you.
You have to include it as an option in your thinking in order to gain clarity.
For example, although it can be insecure, it permits the possibility of moving somewhere cheaper and keeping your existing job and networks.
It also permits a choice about which places you might consider moving if you were to move out of London.
(People who are reading this as you dissing other cities as a choice are really, really missing the point. The point is that you lose an ability to choose if you go for the secure accommodation. And humans get quite anxious when they are faced with uncertainty and lack of choice.)
It's a gamble and a risk.
From the outside, all I can say is that you are being too either/or about the wrong issues and need to break your thinking down a bit.
Secure accommodation is not secure if it can't be a home.
Honestly, posters on MN seem to go completely mad about any thread with 'London' in the title. You'd have had more thoughtful replies - which had actually read your OP) if you'd posted for advice about choosing between two pairs of shoes.
(And apologies there to the posters who did reply thoughtfully. I know you did. It's the loons who go completely la la on any London thread ... I'm past the point of finding them endearing.)