Very busy afternoon, here. But I will dash this off and then return to more fully explain.
Please notice...you did not address free will. You addressed "choosing". If God manages your choices the way He described through Solomon (The king's heart is like channels of water in the hand of the Lord; He turns it wherever He wishes, Prov. 21:1) then it is "free will" that is in question, not choosing.
Free will is defined (universally, I believe) as the ability to select among a group of options with no outside influence exerted upon the man. Correct? More later.
Actually, that is not universal. It is the Nominalist idea of free will. Another view, held by those some call Scholastic Realists, is the free will is the ability to chose the good and to act on the choice.
It is one of the most interesting discussions in the whole thing. It's a shame that you probably won't want to read it, but Servais Pinckaers, O.P., The Sources of Christian Ethics, one of the most challenging books I've read recently presents an exhaustive, even magisterial discussion of this question.
“Free will is defined (universally, I believe) as the ability to select among a group of options with no outside influence exerted upon the man. Correct?”
Not even close. This is a discussion of free will vs predestination, and specifically sparked by my comments that I reject the L & I in TULIP.
L = Limited Atonement. Did Jesus just die for a prechosen few, the elect? Or did he die for all?
I = Irresistible grace. Does God, having chosen us as individuals, give us new hearts and then make us believe? Or did God choose to save those who believe, and cause us to come alive through our faith in Him?
If I have incorrectly defined the L & I, I stand ready for correction. Perhaps some define Limited Atonement and Irresistible grace differently. I’m a retired military officer, not theologian. I’ve never taken a class in philosophy, nor theology, with the possible exception of the Systematic Theology text our Sunday School class has started working through - and we haven’t gotten to the PD/FW chapters.
So correct away, and let us debate without the confusion of differing terms. We agree that scripture says we do not seek God, and that was certainly true in my case. God seeks us...but does he give us the choice of believing, or not?
And if I have utterly misunderstood the teachings of Reformed Calvinists...I apologize in advance.