Getting a flat to rent on benefits by New year

Reaching out because I'm not really sure what to do.
I'm autistic and in my late 30's, I live with my brother and he's evicted me (mental health/possible substance abuse).
He told me I need to be out by the end of the year and he doesn't appear to care what happens to me. I have a lovely sister but she has limited time to help and can't do this for me. I have no other family.
I have some savings but I can't work and have no guarantor. 
I'm over 35 so I think I'll be eligible for enough to rent a flat, I don't currently get rent money because I live with my brother - I pay it out of the Universal credit I get because I am unable to work. 
I've applied for PIP but only sent it off on Monday.

I don't know how to approach this. I will have the rent money but I don't know how to get the flat. If I become homeless I can temporarily stay with friends but I don't know how I will cope with this because of my autism. I also don't want to be a burden to anyone. 
Any advice would be much appreciated, thanks.