That's a tough situation. Assuming you are on a month-to-month basis and not on a current lease which expires in more than 60 days, the landlord has the absolute right to serve a 60 day notice, without cause. You can ask and/or try to negotiate for more time, but the landlord has no obligation under the law to work with you.
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It would be up to the landlord, but he/she is not required to. If the landlord has to initiate eviction proceedings against you it can may take 30 days or more to get a sheriff to perform a lockout. With that in mind, you may try to negotiate a 90 move out which the landlord may find appealing to save the court fees. It's unlikely that you would get much more time than that.
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