avoiding being moved onto UC

Hi! I am in a complex situation , being forced to move house, following section 21 notice, also desperate NOT to be re-assessed right now, due to medical cock-ups discussed elsewhere. Am on ESA, PIP amnd housing benefit

My understanding is that a move onto UC should NOT entail a re-assessment, vbut that it often happens anyway, and cannot be stopped, once set in motion,

Also, I looked into this very carefully and found that if i stayed within the same local authority, then my housing benenfit claim would be considered continuous , so would not be migrated onto UC

All well and good, but with jast three weeks to go, I have faied to find acccomadation withing this LA, but se my chances are much betterm, and rent is also cheaper in some neighbouring towns, but that would certainly entail a change of LA

I also note that various changes have occured since i did mmy research. But I am now to mucvh up to my ears in it to repeat that research.

Is it still true that I shall have to stay within the current LA to avoid being moved onto UC? But then that might well enrtail becoming homeless, whic I iimagine wouild also trigger big changes in benefits...assuming that I still care by that stage of the game! ( I don't think o shall survive being homeless very well)

Anyone have up-to-date info on these issues?