No proof? Did you pay by cash without any documentation of payment?
Go where? And why can't you just go with him to wherever he's going?
Is he all packed and leaving tomorrow?
What's the problem with needing more time? I've had tenants move out over night with family, children, pets, and furniture. Many apartment buildings with vacancies can have you moved in the day you sign up.
Bottom line: If BF goes tomorrow and you stay, nobody can predict what the LL will do. She might be able to get the police to put you out as a trespasser which won't give you any more time or she might have to file against you in court which will give you more time.

