First up: whatever we come to understand about salvation and election, we should be aware that the common rhetoric of evangelicals is surely wrong on this. Evangelical theologians may make the right distinctions, and evangelicals who ponder these things over time may come to get it, but the language used in revival/camp meeting/street evangelism falls into the heresy of Semi-Pelagianism, and even Pelagianism itself. They insist that your act of choosing is the determining factor in one's salvation. Choosing is, if you will, a "work" that one must perform to be saved.* As I noted, evangelicals who read the Bible seriously do note the times Jesus states that they did not choose but have been chosen, and enter the long, messy discussion about what, precisely, is occurring in this interaction between God and man.
I won't give my opinion on the matter because I no longer know what it is. And it's off-topic anyway.
I would like to set the stage for talking about choice in the NT but setting out two OT stories for your contemplation. In Exodus 7-11 God talks about Pharaoh's heart hardening. I wouldn't rest an entire doctrine on it, but there does seem to be a switch in the middle. Up until chapter 9, the scriptures say that Pharaoh hardened his heart. After that, they say that God hardened it. As God says in chapter 4 that He will harden it, the change may be unimportant. And second, the story of Naaman, in 2 Kings 5, healed by Elisha by washing in the Jordan. He almost doesn't do it, being insulted that Elisha did not come down to him. But his servants talk him into it by repackaging Elisha's words so that they don't carry any insult. So Naaman nearly missed healing because of being unwilling to listen.
Moving to the NT, particularly Matthew and John. The whole concept of Jesus preaching, teaching, and explaining seems incompatible with any idea that the people he was speaking to have no choice, or no ability to choose him. Yet there are many verses which suggest exactly that. The end of John 5, and in the positive, John 15. Matthew 17, perhaps.
There is in fact a range of description in the gospels about how much choice we have, and I think this is intentional. But the bulk of them seem to fall into two strongly-related possibilities: 1) The choice has already been made by the hearers, a product of many choices over the years which they cannot now undo, and 2) This is their last chance. All the signs and miracles, even the most primitive one of "He gave us food," which Jesus criticises as been even beneath "He did miracles" as a reason - all these are brought out in a final attempt to get their attention. The speaking in parables - a way to get behind people's defenses and perhaps see what applies to them but they deny - are a last-ditch strategy to slap them to their senses. We moderns think of parables, and miracles, and rescues as charming introductions to the idea of Jesus - and I see little harm in using them that way in the instruction of children and new believers. But Jesus seems to say the opposite: you have had many chances. The result of your choices is now being revealed. In the early chapters of Matthew He says "you must change, you must grow, you must turn." By Chapter 7 He is already saying "I sense some are rejecting me," which progresses to "Some have rejected me," and "There will be very few." By Chapter 21 He is telling the frightening parable of the vineyard.
Peter's affirmations in Matthew 16 are described as being not quite his, but from God the Father and from Satan. Yet Peter's choice is hidden in there somewhere. We think of Christianity as affirming the principle of human agency, of choice, and leading to ideas of freedom and the value of the individual, and of freedom. And so it does, but perhaps only in comparison to everything else, because the NT certainly provides no ringing endorsement of how free we are to make moral choices at any given moment.
*My favorite on this was the old Baptist preacher who explained God votes for you, the Devil votes against you, and you cast the deciding vote. It's discouraging to think we still have to refute Zoroastrianism, let alone Pelagianism, at this late date.