Similarly, a lost person has choice, but he is not free to choose to do good in God's eyes. That is outside his nature. Once God changes his nature, through salvation, then he is free to do good.
It still seems to me you are saying only saved Christians can choose to do good.
Is this a correct stating of your view?
Yes, only saved Christians can choose to do good in God's eyes. Lost people can choose to do good in our eyes, but not God's.
Let's say I am a lost person, and on Monday I decide to help a little old lady across the street. Then on Tuesday, I meet Harley, and he lovingly preaches the Gospel of Christ to me. I am immediately convicted and fall to my knees, a broken man. I earnestly pray to God to forgive my sins and ask Jesus Christ to come into my life as Lord and Savior. I then turn to Harley and say: "Thank you Harley for leading me to Christ, without what you did for me I might have died a lost man". Harley then slaps me upside the head and asks if I have been listening to anything he has told me. :) Anyway, then on Wednesday, I decide to help the same old lady across the street again.
Now, I would say that on Monday I did nothing pleasing to God, but on Wednesday I did. A major difference is that on Monday, I did not help in order to glorify God, but on Wednesday I did. We as outsiders would not notice the difference, but to God there would be a huge difference. Slaves to sin (the lost) cannot do anything because Christ is not in them. I think this is the sense of:
John 15:5 : "I am the vine; you are the branches. If a man remains in me and I in him, he will bear much fruit; apart from me you can do nothing.