I had always taken that to be talking about the flesh body as well. Is it possible it is also speaking of the soul-life? I don't know, maybe.
I also think it was the flesh life. The life is decaying as we start to live in christ who lives in us.. As he grows us or sanctifies us.
I also see this outer man that is decaying as the flesh (sarx), meaning our bodies (soma).
I see the inner man as being our new creation spirit. That which is born of Spirit is spirit. We are the spirit children of God.
I see the soul as being the same as "psuche", meaning our self-identity, our thoughts and feelings, the ongoing mental "chatter" in our minds.
I see me as being born in a flesh body, with a flesh brain, that contains "me". The flesh is corrupted by sin, which means it's damaged, and will only produce a mind that reflects that damage, the corrupt mind of the flesh.
I live and grow, and experience things, and make choices, and all the sinful things around me, and that I choose, all leave their imprint of more corruption. Sinful thoughts, feelings, actions, all leave their imprint in the brain, as well as all the evil done against me. These are the neural pathways, "reward circuit", the biology of our minds. Traumatic experiences in developing children, and even adults, can leave permanent damage to the brain, impacting it's functionality.
So my body, my brain, my mind, all are corrupted by sin, all are dead in transgression and sin, all is condemned by God.
And then I am regenerated.
Now there is a new part to "me", a living spirit, united to God's Holy Spirit. This new part, my spirit, does not live in my brain, like my soul does. It lives in the celestial realm with God.
My consciousness is in my soul, psuche. My psuche was determined before only by the corruption I labored under, and whatever I've been able to do working with that. But whatever I was arose from corruption.
Now my psuche has another influence, 2 actually, my new spirit, and the Holy Spirit. These work together as one.
Flesh and spirit war against each other for control of the soul, psuche, and subsequently control of the body, that is, our actions.
Being crucified with Christ is when "I", my self-identy, sense of self, and source of self, is cut away from the flesh/body, and is transfered into Christ. I am no longer the man of flesh in Adam, I am now the man of spirit in Christ. But my soul is still in the body, and "I" still see from the body's eyes, and walk with the body's feet, and so on.
I understand walking in the Spirit to refer to living entirely in that new man, that is, that the new man, the spirit child of God, the one who is united to the Holy Spirit, entirely overwhelms any flesh influence or consideration. This is when the soul/psuche becomes completely aligned with the Holy Spirit, and the flesh is competely controlled.
I think the common experience among the regenerate is that we are in the process of being trained by God to recognize and subdue when what is going on in our minds has it's source as the flesh, so we can stop that, and replace it with thoughts/feelings/choices that have their source in the spirit. I think this is what is meant in putting off the old man, and putting on the new man.
This process of training can be very extreme sometimes.
In my case, I've had a very corrupting life experience, and have had very enduring maldevelopment, and deeply ingrained bad habits of how to think, and how to feel, and how to choose. And these are the fleshy lusts that war against the soul.
And when I cannot get a handle on things, when my present ability to choose spirit over flesh fails me, God steps in with His chastening. I see His chastening as very difficult and often painful experiences that overpower the working of flesh against me, like a hammer that breaks a chain. Something so overwhelming that it causes me to reject what I was holding on to, and to accept His righteous way. Putting off, and putting on, with a heavy hand to get me there.
Spiritual maturing is the culmination of these things, as we learn to identify fleshy thoughts/feeling/choices, to deny them before they come to any fruition. In the denying of the fleshy stuff, this allows the spiritual to grow unhindered.
The more we think/feel/choose according the Holy Spirit, the more this reworks our soul/psuche into the spiritual pattern, away from the fleshy pattern. And this enhances are ability to think,feel/choose in the right way.
If we are walking according to the flesh, this is to live as one not regenerated.
If we are led of the Spirit, this is to be making our choices of what to think and feel and do according to what the Spirit whispers to us.
If we are walking in the Spirit, this is to be living according the Holy Spirit in us, without regard towards our flesh. Again, I only say these things to show how I understand this.
I think the common Christian experience is to find ourselves in each of these states from time to time. I think this will depend a lot on what we think is happening in us, and what our faith is.
My faith is that in Christ, I'm fully separated from my flesh, that's no longer me. I'm just using the body. My faith is that God fully empowers me apart from my body, which is to be my tool, not my source. My faith is that God WILL complete His work in my, which will result in transforming my body into one that will no longer war against me with it's own ungodly lusts.
I think that when we walk in the Spirit, the Holy Spirit's fruit, Love, joy, peace, et. al., is our life experience. Not that we grow into being more loving, rather, our choices are all choosing love. Our feelings are joyful, peaceful, even in the midst of suffering and pain. Paul and Silas singing in jail.
I think when we are led of the Spirit, we are building a better character with every godly thought/feeling/choice/action, as this writes different neural pathways into our brains. This leads to less resistance from our old man, as the mind is being renewed to the new man. As the soul is being rewritten according to the spirit.
This is where we grow to be more loving, our character improving.
And of course when we completely lapse in our faith and revert to "walking according to flesh", living as if unregenerate.
For me, the battle is in the mind. The sooner I recognize fleshy thoughts, the sooner I can recover control. But best of all is that I'm learning the faith that sets all that aside and allows me to walk in the Spirit.
Much love!