First, I am a Christain. After accepting Christ, I tried running away from anything that could bring me down spiritually. I was told I was practicing self-righteousness. I lived my life and I fell into sin, but when I fall there are these feelings of guilt in me, and I go to God for forgiveness. After some time where it happens repeatedly, I notice that when I sin, I don't feel guilty any longer, even though I know I have sinned. It's scaring me. Please, what's happening?
Second, is the statement right that once you have accepted Christ you are not a sinner, even when you sin, it won't be counted against you because you have an advocate in heaven?
Becoming a Christian is more that accepting Christ. See: What Saves a Person?
You are supposed to stay away from sin. "But examine everything carefully; hold fast to that which is good; abstain from every form of evil" (I Thessalonians 5:21-22). It would be self-righteousness if you were defining what is good or evil. If you are obey what God says is righteous, then it is a righteousness not of your own. "For I testify about them that they have a zeal for God, but not in accordance with knowledge. For not knowing about God's righteousness and seeking to establish their own, they did not subject themselves to the righteousness of God" (Romans 10:2-3).
Some people misread Paul's other statement on this topic: "More than that, I count all things to be loss in view of the surpassing value of knowing Christ Jesus my Lord, for whom I have suffered the loss of all things, and count them but rubbish so that I may gain Christ, and may be found in Him, not having a righteousness of my own derived from the Law, but that which is through faith in Christ, the righteousness which comes from God on the basis of faith, that I may know Him and the power of His resurrection and the fellowship of His sufferings, being conformed to His death; in order that I may attain to the resurrection from the dead" (Philippians 3:8-11). They see "Law" and incorrectly assume that Paul is talking about all law, but he is really talking about the Old Testament law. And notice that it wasn't the law itself, but a righteousness derived from the Law. This is what the Pharisees did. They imposed rules based on the Law, but not found in the Law, and then claimed that people had to follow their rules too (Matthew 15:1-9). But faith demands that we live by "the righteousness which comes from God."
"Do not enter the path of the wicked And do not proceed in the way of evil men. Avoid it, do not pass by it; Turn away from it and pass on" (Proverb 4:14-15).
The problem when you sin repeatedly is that your conscience stops telling you that you are doing wrong. "But the Spirit explicitly says that in later times some will fall away from the faith, paying attention to deceitful spirits and doctrines of demons, by means of the hypocrisy of liars seared in their own conscience as with a branding iron" (I Timothy 4:1-2). That is what happens when you burn yourself repeatedly. The scar tissue no longer has the sensitivity that it once had. Thus, the answer is the same that Paul gave: "What shall we say then? Are we to continue in sin so that grace may increase? May it never be! How shall we who died to sin still live in it?" (Romans 6:1-2).
Thus, your second question is not exactly correct. A Christian should not sin, but sometimes he messes things up. In such times, Jesus gives us a way back out of sin.
"This is the message which we have heard from Him and declare to you, that God is light and in Him is no darkness at all. If we say that we have fellowship with Him, and walk in darkness, we lie and do not practice the truth. But if we walk in the light as He is in the light, we have fellowship with one another, and the blood of Jesus Christ His Son cleanses us from all sin. If we say that we have no sin, we deceive ourselves, and the truth is not in us. If we confess our sins, He is faithful and just to forgive us our sins and to cleanse us from all unrighteousness. If we say that we have not sinned, we make Him a liar, and His word is not in us. My little children, these things I write to you, so that you may not sin. And if anyone sins, we have an Advocate with the Father, Jesus Christ the righteous" (I John 1:5-2:1).
Notice though that a person has to want out of his sins and asks God for help and forgiveness. No faithful Christian will accept sin being in his life. He'll do something about it. The statement you gave makes it appear that there is forgiveness without effort. See: Is "Once Saved -- Always Saved?" a Bible Doctrine?