In chapter 1 of Romans, it says that men are without excuse – God can be seen in nature. Aren’t we all WITH excuse if He is the one Who chooses to reveal Himself to whomever He desires?
Here again, we can go to Romans 9 (see vs. 19). But God’s sovereignty does not compete with man’s responsibility to turn to him. Somehow, and I don’t now how this works, God calls—or not—and we are responsible to come to him, even though we can’t come unless he draws us (John 6:44).
But that is how it is everywhere in the Bible: God causes and men choose. In Genesis 50:44 we see that Joseph was sold into slavery by his brothers who meant it for evil (because they hated Joseph), but God sold him into slavery for good (to protect and preserve Israel through famine). We see the king of Assyria being used by God to destroy sinful Israel, but the Assyrians undoubtedly thought they were unbeatable, for which later they were destroyed (see Isaiah 10:5; 12). Then in the New Testament the Jews killed Christ, but it was in accord with the plan of God (Acts 4:27-28). This same kind of thing is shown to be the norm all through the Scriptures, these are just some of the more obvious cases.
So, in the end, God predestines and men are held responsible for all of their choices.
A couple of good books on this topic would be The Reformed Doctrine of Predestination (read the whole book) by Lorraine Boetner and The Sovereignty of God and Human Responsibility by D.A. Carson.
I hope this helps,