Donald Campfield :
I subscribe to 'TULIP' because I can see all of those things taught in various Bible passages. That said, there's a fuller explanation that can, and probably should be given. Re: predestination vs. free will—and this may sound like a cop-out, but I don't intend that—I think that BOTH of those things are operative in reality, though our human frame-of-reference can't perceive them in the same way that God does. From the human perspective, we all make apparently free choices every day of our lives, and most of those involve some degree of weighing the options and choosing among them. But there are inherent limits on human decision-making as well, which we sometimes don't consider. I can't choose to travel to the Moon next week, though some humans have done that in the past. The option to do so requires resources and circumstances that lie outside my personal control. I can't decide to change the current President of the United States for another person tomorrow, just because I'd prefer the other person; I live within a society that has laws and rules in place governing who is President, and when and why, and those restrict me from making my preference a reality. Likewise, I cannot defy natural law at will, so that I can suddenly negate gravity and fly using my arms. I have to enter an aerial vehicle that's designed to behave IN ACCORDANCE WITH natural law in order to fly. And before the Wright brothers and hot air or lighter-than-air balloons NO ONE could choose to fly using those means. So my free will is operative up to a point, but is limited by the available resources and by various circumstances that lie beyond my personal control at a given point in time as well.
From the Divine perspective, everything which happens is apparently pre-arranged—somehow and on some level(s). God fulfills prophecy not just via His omniscience but also by His control and guidance of history (I understand the Bible to be telling us). And my understanding is also that God's predestination encompasses all human activity, and all angelic activity too for that matter, so that the portions of our lives where we perceive and engage in our 'free' will are all a subset of God's larger plans and comprehensive control at the same time. But we're not told much more in scripture than that about how God manages such control overall, only that He does and that he does it righteously, so that no one will be able to complain that God ever treated them unfairly or unjustly. But, again, on the human level, and from the human perspective, we do have free will—but only within certain boundaries, and it's on that level that our free choices, which include our sinful choices, render us culpable before God for them.
So that's my best quick thumbnail-sketch of how I think predestination and free will are both true at the same time, and fit together. Human free will is not absolute, though God's decretive will (i.e., what actually will happen) is, and the two overlap and yet are both true at the same time. And we all on planet Earth are HIS creatures, living in HIS universe, which is governed by HIS natural and moral laws, and then we are further limited by our available resources and our personal circumstances. But after all of that is granted first, our will is completely free!
LIke you, I also believe that all humans CAN be saved and that Christ's atonement was sufficient for ANY number of repentant sinners. So far as I know, Calvinism of the kind I see taught in scripture does not contradict or override all the other passages which instruct Christians to evangelize, preach the gospel, and otherwise fulfill the Great Commission. John 3:16 tells us "whoever" believes in Him, etc. Where Calvinists and nonCalvinists sometimes diverge is over a misunderstanding of who the gospel is offered to (all) vs. who will actually respond to it positively and be saved (the elect). Scripture tells us not everyone WILL be saved but it also tells us anyone and everyone COULD be saved by Christ's atonement. So I evangelize heartily as a Calvinist, and have known many others who do the same. You might be interested to know that an American Presbyterian evangelist of Scottish background—named Robert Manderson—once put the matter of God's predestination not being in actual conflict with human free will this way: "God never makes you go against your will, but He sure makes you willing to go!" (Bob himself had resisted being saved for a long time before becoming a believer, but his family prayed for God to do ANYTHING to get him to consider the gospel, and their prayers were duly answered—Bob had all manner of calamity befall him, which then led him over time to salvation in Christ).
So like you, I also believe that anyone can be saved if they meet the condition of repentance and faith in Christ (i.e., they embrace the biblical gospel). In practical terms—because we do encounter terminology such as "predestination" in the Bible's teaching—I think it's essential to recognize that we are still commanded to carry out the Great Commission and to evangelize 'indiscriminately' because God has not 'published in advance the list' of those who won't be saved. HE knows who they are, and HE knows who will be saved as well, but humans do not. So we rightly preach the gospel to everyone, warning of the wrath to come in Judgment Day. Some will listen and be saved; others will not listen, or will listen but refuse. Christ's atonement is sufficient for any/all, but will ultimately only have been 'applied' to those who are saved (by definition, really).
I hope my explanation is at least a little satisfying. It doesn't get into the larger question some people have—Doesn't God predestine many people to hell?—and my answer to that sort of 'fairness' question is Yes/No. It's "Yes" because God controls all things, but it's also "No" because God allows enough 'elbow room' that those who reject the gospel do so freely and so are culpable. NO ONE is turned away who truly WANTS to be saved. People end up in hell because they exercised their free will to reject the gospel. And at that point we're pretty much exactly in Romans 9:19–24.