I always send a email the end of the same day I send in an application to check if they have received it and introduce myself . I them follow up in 3 days to see if they need anything else form me and to see how it's going .

Def follow up and give them a call , at least you will know if you need to start looking more urgently .

How long do you have left?
I know it's not want you want to hear but you may need to start applying for things you don't really want , anything will be better than having no home at all

Hugs Iv been there xo