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.
Please note that no attorney-client relationship is created or implied by the posting of this answer.
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.
By posting on this site or answering/responding to questions does not create an attorney-client relationship and is intended to be an opinion only. My opinion is not intended to be a guarantee or promise of any outcome or result in your matter.