What's the Penalty for Holding Over

My question involves landlord-tenant law in the State of: minneasota
My lease is up August 31 and I had given the landlord 60days notice. There is a clause in my lease saying that month to month is an option except my landlord didnt wanna do month to month. I am willing to sign full lease now except they said the spot already taken and I cant find anything w such short notice.
1. If I dont move out by the end of the month aka when the lease expires, would they evict me? if so how long/lengthy the eviction will take? or will I just loose my sercuirity deposit? Would they be able to just throw my stuff on on that day?
2. Since month-to-month option was written & they're not open for discussion for that option, anything I can do about that?
3. they told my my place was already rented out, how would this affect me?
thanks

It's up to your landlord to decide what steps to take if you hold over. Obviously if you put the landlord at risk of losing a new tenant, it's reasonable to assume that your landlord will be inclined to evict you as quickly as possible. Eviction is a legal process.

You are free to try to negotiate with your landlord about staying longer, but as you've indicated that the apartment has been rented to somebody else it's reasonable to assume that your landlord will continue to say "no".