Your are correct in understanding that your landlord needs to give you the 60 day notice in writing. If you need more time to find a place you can claim the notice is defective. His defective notice likely does not relieve you of your duty to give 30 days notice that you are going to vacate.
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Your landlord's verbal request is not a legal eviction notice. I would not assume that was a legal eviction notice and I recommend that you give your landlord 30 days written notice if you plan to move out.
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