Couldn't it simply mean that the dying process started at that point? In other words, all the millions of cells that make up the human body. All the organs and tissues and muscles that make up the body. Before they weren't dying but were immortal. But, when sin entered into the world so did death because the wages of sin is death.
Now just imagine for a second that Adam and Eve didn't sin. Eve said no to the serpent, they multiplied as per God's command on the 6th day, and replenished the earth. Death just didn't happen.
If we've got our thinking caps on, do a little math. Adam and Eve would still be with us and the earth would be filled trillions upon trillions of people.
So as a believer in immortality (of the body and soul) from the beginning - which coincides with Saul/Paul's take on death and sin, one sort of has to accept that either the earth was going to be overpopulated in short order if death didn't happen, OR that "death and sin" were part of God's plan.
Or
The assumption of bodily immortality of Adam and Eve is wrong. Therefore, if physical death isn't being spoken of, then it's something else.
John 8:51 Verily, verily, I say unto you, If a man keep my saying, he shall never see death.
Was Christ wrong? After all He also taught He would die and did in fact die.
Edit: Post thought. In Genesis 6:3 God reduces how long man will live. The shortening of life span from hundreds of years to 120 years mentions nothing about dying sooner due to some sort of inherited sin nature.
Edit2: Infants who die of sudden death syndrome or some other reason also must have died with some relation to Adam and Eve's sin, right? While mass murderers can live 50,60 70 years or more. That's what follows when you believe Saul/Paul!
Lovely.