Only if you believe in irresistible grace does that false dichotomy exist. God is sovereign, man is responsible. Man cannot come to Christ unless he is drawn, which simply means to "impel," bring to a place where one can be taught, where one will listen, or at the very least be held. That isn't impelling to belief, that is simply drawing to a point where one can hear what is to be said. Man cannot respond to the call of Holy Spirit without the Spirit opening his mind. But his responsibility enters in at that point. He has the ability to reject the truth revealed to him, and many do.
In John 3:1-10, we have as clear a presentation of God's sovereignty as there is in the Bible. Immediately on it's heels, in vv. 11-21, is as clear a presentation as exists in Scripture of man's responsibility. If man has a responsibility, then he must have some part in the decision to follow Christ, even if it is so small a part as believing what only the Holy Spirit can enable. God does not manually insert the Holy Spirit into the human brain. He must be invited in.
In the John 3 passages cited, Jesus speaks to Nicodemus, at this point a doubter, a non-believer, though he is obviously seeking to understand the truth of Who he thinks Jesus might be. And Jesus immediately introduces to him to twin and parallel truths of divine sovereignty in salvation and human responsibility to accept it. In essense, Jesus says, "This is a work of God, solely a work of God, but you will be held responsible if you do not believe, and you are called to believe and eternal life awaits you if you will believe." Those are twin truths that run parallel.
How do you get these two truths to merge? You can't. They will always run parallel. They will always run parallel. They will never come together. They will never intersect. They will never be diminished. Legitimately, they are what they are. The fact that you dont understand how they go together only proves that youre less than you should be, and since I don't understand them either, so am I. It doesnt say anything about God. Our inability to harmonize those things is a reflection of our fallenness. When someone asks me to harmonize these biblical facts, these truths, I have to say, "I cant."
They cant be harmonized in the human mind. But realize this, we are puny minds. It is an utter waste of time to try to reconcile these truths. They can't be reconciled in the human mind. It is impossible. Yet Calvinism pretends to do so. That's nothing more than man's folly and insistence that he not be denied understanding. Rather than simply accept God's truth, no matter how impossible it seems, we invent a God who is capricious and cruel, claiming He created some who can never know Him. How arrogant!!
Collectively we are puny compared to the infinite, vast, limitless mind of God. All I can tell you is that in the Word of God, these truths run parallel. And the answer is to believe them both with all your heart. The one, divine sovereignty, will inform your worship. The other, human responsibility, will motivate your evangelism. Why seek to do the impossible? Just believe. If we are ever intended to understand, we will -- in heaven.