If we are born with a sinful nature (an inclination to sin) from Adam, does that really mean we have free will, since the sin nature makes us more likely to choose to sin?
When the Bible speaks of our sinful nature, it does not just mean "an inclination to sin". It's not just a likelihood of making a sinful choice that is the result of having Adam as our original forebear as the federal 'head' of all humanity. All humans WILL make sinful choices, even when they make apparently good choices. The Bible says,
"All of us have become like one who is unclean, and all our righteous acts are like filthy rags; we all shrivel up like a leaf, and like the wind our sins sweep us away" (Isaiah 64:6).
That is partly because even an apparently good choice could have sinful, ulterior motives at back of them, or (due to lack of information) the choice turns out to have very harmful results either to one's self, or to others. The motive may have been good, but even so, some choices result in sin further down the line.
We all have responsibility to choose one way or another, but responsibility to choose does not equate with “free will”. Not until the Son has truly set us free can we first see that all our choices up till then were made on the basis of being enslaved – to sin. Jesus spoke of true true freedom being freedom from our slavery to sin. If you study John chapter 8 verses 23 to 47 you should see what is required for sinful humans to start becoming truly free. In verse 34 he said:
"I tell you the truth, everyone who sins is a slave to sin. Now a slave has no permanent place in the family, but a son belongs to it for ever. So if the Son sets you free, you will be free indeed."
Christ first has to liberate the captives who truly believe in him, and trust in him alone to be saved, as opposed to still holding on to their will to make choices. Such people have fallen between two stools. Those who have been given true freedom as a gift of grace, have been picked up from the ground (as it were) and seated in a heavenly ‘place’ even before they get to heaven. From heaven’s vantage-point we then see clearly just what the matter of choice, responsibility to choose even while sinners, and freedom to choose the will of God, in Christ, means.
Once people have been adopted into God's heavenly family (by the faith of Christ) then they have their permanent place in that heavenly family and are no longer slaves to sin. They will still continue to make sinful choices but as they grow in faith, they will desire more and more to do only what pleases their heavenly Father, and as they learn about Christ, they will know the Holy Spirit convicting them of the extent of their sin nature, and abhor it more and more. They will turn from sin, going in the opposite direction. Only then will it dawn on them just how enslaved they were to sin prior to being born again by the Holy Spirit.
This is a huge subject but until a person has been convicted of their sin by the Holy Spirit, they are never going to understand the depths of meaning behind what Jesus did to liberate sinners from their slavery to sin. Only once that liberation happens can people really begin to discover what free will is all about.
EDIT: I have answered from a Protestant, Reformed Presbyterian stance and, as others have pointed out, you need to hone your question if it is to be reopened; you need to ask for a particular response from particular groups, and not leave it wide open to all and any who think to give a Christian answer, in general.