"But why then Paul is using the present tense "I am" as if he was still a sinner at the time of the writing of his letter?"
To help answer your question here is a literal translation of that verse.
The word [is] steadfast, and worthy of all acceptance, that Christ Jesus came into the world to save sinners—of whom I am first;
The word first means;
4413 prṓtos (an adjective, derived from 4253 /pró, "before, forward") – first (foremost). 4413 /prṓtos ("first, foremost") is the superlative form of 4253 /pró ("before") meaning "what comes first" (is "number one").
The main point of this passage is that Jesus Christ came in the world to save sinners. Since he came in the world to save sinners He wants to show forth or display His long suffering And as a pattern for those who are about to be believing on him for life of the ages.
12And I give thanks to him who enabled me — Christ Jesus our Lord — that he did reckon me stedfast, having put [me] to the ministration, 13who before was speaking evil, and persecuting, and insulting, but I found kindness, because, being ignorant, I did [it] in unbelief, 14and exceedingly abound did the grace of our Lord, with faith and love that [is] in Christ Jesus: 15stedfast [is] the word, and of all acceptation worthy, that Christ Jesus came to the world to save sinners — first of whom I am; 16but because of this I found kindness, that in me first Jesus Christ might shew forth all long-suffering, for a pattern of those about to believe on him to life age-during: 17and to the King of the ages, the incorruptible, invisible, only wise God, [is] honour and glory — to the ages of the ages! Amen. YLT 1 Tim: 1:12-17
This is something new that was being proclaimed from the glorified risen Christ.
Previously: in contrast;
('I was not sent except to the lost sheep of the house of Israel.' Matthew 15:24)
Paul says he was the first, The number one example of this grace as being displayed in him first. This grace is being shown from a Glorified Christ outside the land of Israel.
4413 prṓtos (an adjective, derived from 4253 /pró, "before, forward") – first (foremost). 4413 /prṓtos ("first, foremost") is the superlative form of 4253 /pró ("before") meaning "what comes first"
Jesus Christ is going to show what it means to actually save someone who certainly hates him and wants no part of him. It is His life inside a person that totally transforms someone by his resurrected life inside them.
13For you have heard of my former way of life in Judaism, how severely I persecuted the church of God and tried to destroy it. 14I was advancing in Judaism beyond many of my contemporaries and was extremely zealous for the traditions of my fathers.
15But when God, who set me apart from my mother’s womb and called me by His grace, was pleased 16to reveal His Son in me so that I might preach Him among the Gentiles, Galatians 1:16
Paul is His first example of a new type of grace that is being proclaimed in the earth.
He goes after someone who is making havoc against His own brand new believers who doing horrible things to them. This man Saul is not at all in the state of repentance, He did not believe in Jesus Christ, And he certainly was not showing any good works at all.
He does not kill this man but instead overwhelms this man with his own grace and faith and love…
Paul is used as a backdrop to reveal the patience of Christ. He needed someone like Paul to show the incredible patience of Christ towards this man who was persecuting His believers, putting them in prison, much hatred was being expressed through his grievous acts.
makrothumia: patience, long-suffering
Definition: patience, long-suffering
Usage: patience, forbearance, longsuffering.
3115 makrothymía (from 3117 /makrós, "long" and 2372 /thymós, "passion, anger") – properly, long-passion, i.e. waiting sufficient time before expressing anger. This avoids the premature use of force (retribution) that rises out of improper anger (a personal reaction)
It is the greatness of Christ that is revealed making this man into someone new by putting his own life inside a man that's just the opposite of him.
Paul is using himself as the first example of Christ patients towards him. He claimed that spot at that time so anyone who became a believer after him would know the incredible patience of Christ.
Christ indeed saved Saul and now see Christ life revealed through Paul. It was the love, grace and faith of Christ that saved someone to the common eye of someone unredeemable because of everything he was and did.
“It pleased God . . . to reveal His Son in me” (Gal. 1:15-16).
Paul holds a unique place in the revelation of God to man, as the one who was chosen of God to reveal the mystery of Christ:
Paul could identify himself as the chief of sinners and was keenly aware of the grace that had been extended to him.
It is because of him being the foremost of sinners that makes him a witness to testify to the gospel of the grace of God. Acts 20:24