My view - I am new here so take it easy ok! :-)

The nature of sin is dead - crucified. Paul does not say it is partially dead, it has been crucified with Christ.
But, our bodies are still alive and still remember the things it used to enjoy doing according to the old sin nature and that is what plagues us. Our brains are just muscles and stay behind and rot when we die and it is the memory banks that retain the old natures memories.

The suffering we partake of with Christ is to treat our bodies (body of sin, from the man of sin) desires and passions as dead (put to death in Colossians 3 is not a good rendering) and to rather seek Christ who is seated at the right hand of God.

Does this make sense?