To my experience the "cord" part is not always there and it is not some sort of umbilical cord between the soul and the body. You will not die if it is severed. Generally, you'll just feel pretty messed up.
I have read more of the astral cord being used as a "lifeline" within old 2nd edition DnD resources than I have in anything about exploring other realms.
That being said, there are often "cords' of energy that can be observed connecting our spirits to any object, person, or place that we have a strong emotional connection to. This would very much include your body.