I don"t hold with the total depravity view. I believe that man has the free will to chose to accept God's grace and to chose to work with God's grace. St John Cassian (4th Century) who was thought by some to be semi-pelagian, said: "Divine grace is necesary to enable a sinner to return unto God and live, yet man must first, of himself, desire and attempt to choose and obey God" and "Divine grace is indispensable for salvation, but it does not necessarily need to precede a free human choice because despite the weakness of human volition, the will can take the initiative toward God"
I dunno. It seems to me, however, that the Scriptures testify that "no one is righteous, no, not one; no one understands, no one seeks for God," that and that "every good endowment and every perfect gift is from above, coming down from the Father of lights." I take this to say that we mortal beings can't will to choose God and be saved because we are "slaves to sin" before God gives us grace, as St. Paul writes.
I also remember Our Lord's words: "No man can come to me, except the Father which hath sent me draw him: and I will raise him up at the last day." That does not seem to work with the idea we must first seek God.
Let's remember that whatever we think about free will, no party in this discussion is allowed to claim that we can seek God first. That's the heresy of Pelagius, right? Thus even if we have free will, the initial action must be from God first, the 'prevenient grace' (the grace that "comes before")...
I think it is clear that God is the prime mover in our salvation, and it is God who takes the first steps - however we are called to respond. Mary was called by God to participate in God's plan for the salvation of all of human kind. That step is validated in Mary's response 'Behold I am the servant of the Lord, let be to me according to your word'. Each of us confronted by the grace of God must respond, one way or another in words similar to Mary. We need to make her words our own. Then we will understand the words of Augustine of Hippo 'I found God, only to find that God found me'. God initiates and yet we are called to participate in our salvation. - The 39 Articles in my view address this matter fairly well, and the Homily on Justification which I is in the bundle I am getting ready for the online documents addresses the matter further and more thoroughly. There is a balance to strike between relying too much on predestination and feeling no need to do anything about it as the result is already in the bag, and free will where we rely too much on ourselves and not enough on God.
I agree with much of what St. John Cassian taught, he had much to say about living a life of repentance. He was opposed to a one time "confession"of faith, rather he taught that we are predestined to be conformed to the likeness of Christ by living a life of repentance. He and other Church Fathers thought that Augustine wasn't entirely correct with his teachings on predestination/once saved always saved ect. That he hadn't been thoroughly taught Apostolic doctrine on this issue. That's why some Orthodox call Augustine "Blessed"instead of "St."
I have no issue with God's grace coming before. I do think we have free will to choose whether to accept his grace and whether to work with his grace to transform our lives.
I think Wiki does explain the different thought on total depravity quite well. https://en.m.wikipedia.org/wiki/Total_depravity
Then I don't think you have an issue with the traditional doctrine as espoused by the Western Church. The problems you may see arise from certain schismatic mindsets that throughout history have tried to stretch that balance too much to either the side of free will (total independence), or to predestination (total univocal subjection).