I wouldn't cut anything off that's even a bit alive, even though it may be dying...unless, of course, it's infected with some fungi or something. If I were you, I would leave the canes be. Are there any other canes other than the two that are slowly dying? Of course, it's not a good idea to leave a den "caneless".

You should do something about the humidity. With the winter coming and the heat on all day, the air in my home gets very dry and my orchids aren't feeling too comfortable. Humidity is important for orchids...try to do something about it. Humidity trays, misting, etc., all work.