When I moved into my place, my landlord had two copies of the lease, one for myself and one for him. It says on my lease in clear, capital letters "Sign Both Copies Seperately", but my landlord did NOT sign my copy, he only signed his own. Does this nullify the lease, or is it binding because he signed his copy?

No, there is a signed copy that he can easily supply in court. And all it takes is one copy to be binding, the other is merely for your benefit so that if you needed to inforce something he was not doing, you would have proff that they signed it.

So that is not going to get you out of the lease.