I believe the first and second deaths are physical and spiritual. Adam’s spiritual perfection was destroyed when he first sinned. This forever changed his “want to” and made him for all effects spiritually dead. We are born with this spiritual death attached. The only way that any of us is made alive is through the regeneration of the Holy Spirit. That includes Adam. The wages of sin is death, and Adam started paying the moment he sinned. He passed down this debt to all of his progeny and all of us sin. Only Christ can nullify the spiritual death.
The first death needs to come before the second death. Are you saying that was spiritually dead without dying?