I believe there is a lot of confusion over this question and I want to give some scriptural clarity so that we can face this question, not out of a reaction to people who use grace as a license to sin, but rather, out of an understanding of God's word.
What I'm about to write is extremely important to understand. It means life and death, Heaven and Hell, freedom and bondage.
Okay, check out Romans 10, the whole chapter. Paul says that if you confess Jesus as Lord, you will be saved. Now, someone can say, "There are a lot of people trusting in Jesus as Savior, but they are not making Him Lord, and this passage says you need to trust Him as Lord to be saved. So they are not saved. So you have to make Jesus your Lord before He will save you."
Now, let's get into it:
First, you can't simply say believing Jesus is Lord is the same thing as making Him your Lord (making Him number one before all other idols of money, sex, ministry, etc). You have to prove that. I don't think believing something is true and making an effort to live according to that belief is the same thing.
Second, if any verse in the bible means make God the Lord of your life, it is the first of the Ten Commandments: "You shall have no other gods before Me." What else can that mean? God is saying, "Make me number one. If you make anything I created more important than Me, you are in sin." Also, note that this commandment isn't one single action. It's a whole attitude and approach to life that affects all of our actions. Now, Paul starts off Romans 10 saying that the Jews are not submitting to God by seeking to establish righteousness through the Law.
So, if you are seeking to make God number one in your life to be saved, you are under the Law and therefore, you have zeal, but not according to knowledge. You cannot be saved.
Third, "LORD" is a saving name. Saying I trust in Jesus as LORD doesn't mean something different from trusting Him as Savior. Why? In this chapter, Paul quotes Joel, saying all who call on the name of the LORD shall be saved. Why would someone call on the Lord to be saved? It's simple. When Moses said, "Please, let me see Your glory," God said, "I will cause all my goodness to pass before you and declare to you my name, 'the LORD'." And what did He declare? "The LORD, the LORD, a God merciful and gracious, slow to anger...forgiving iniquity...etc."
So, from Moses on, when we see LORD in the bible, we are to think of God's glory. And part of His glory is to be merciful and forgiving. Thus, we call on the LORD to be saved. We don't separate Savior and LORD like water and oil.
What about sin? Two things.
First, see Romans 6. We submit ourselves to God as those who have been saved. We have been brought from death to life. Thus, we have the heart, power and Life to submit ourselves to God.
Second, those who rely on the Law are under a curse and bound by sin. See Galatians 3, Romans 7 and 1 Cor. 15:56.
It's not until you look away from your effort to make Jesus number one before all idols that you can really understand and receive His love for you as a sinner. This love, and the work of the Spirit that comes through faith changes us, sets us free, writes God's laws on our hearts so that our heart's desire is to make Him number one. This is the New Covenant. The Good News. We are saved by grace through faith alone. So that no man may boast.
There will always be people who use grace as a license to sin, but don't react by changing the gospel into something that is not good news at all, that leaves people in the dark knowing they need grace but having no idea how to receive it, trying not to rely on their works and yet feeling they need to make God number one before they will be saved (as though that wouldn't take serious work).
Be discerning. Preach the gospel. Enjoy the gospel. Amen.