Need rent advice

Hi we live in an apartment and lease expires in January. We have to give a 60 day written notice to move out or they automatically switch us month to month. I want to rent a house. I know it’s risky to write a notice without a solid place to move. what if come January we don’t find a house within budget? Currently, I see a bunch out there! But of course they won’t be available in 2 months. What do we do? My fiancé doesn’t want to risk it but I hate my dogs being stuck in our one bedroom. Plus I am going to be moving to working from home indefinitely so we kind of don’t have a choice.