My gut feel would be before as you never know what might come with the birth - you might need a c-section which would put you out of physical action for a while, or you may need a longer stay in hospital etc. I know that sounds paranoid but I guess for me I would consider that there are more unknowns after the birth than before if that makes sense.

What a horrible position to be in - I hope you can find another place easily.

The other thing maybe you could do is ask your landlord (since they are already feeling bad) if they can give you a month rent-free or something so you can take up a lease sooner on a new one and do stuff gradually? Just an idea...