When we come to Christ, the Bible says, God comes to live within us by His Spirit, and we aren't the same people we once were. Our goals change, our habits change, our behavior toward others changes -- all because Christ is now living in us and is the center of our lives.
But this doesn't mean our personalities necessarily become totally different. Before he met Christ, the Apostle Paul (who at that time went by his old name of Saul) was a bold, strong-minded, focused personality with one goal in life: to stamp out the Christian faith. Christ changed this -- but instead of completely changing Saul's personality, God redirected it and channeled it. Saul the persecutor became Paul the preacher, fearlessly carrying the Gospel throughout the Roman world.
Don't misunderstand me, however; God did change Paul. He took the rough edges off his personality, and put within him a new love and joy. And God wants to do the same with you, as you submit yourself to His control. Paul said, "I can do everything through him who gives me strength" (Philippians 4:13).
God knows what you are like; after all, He made you. Commit your independent streak to Him, and ask Him to redirect it and use it for His glory. Ask Him also to take away your rough edges, and make you more like Christ every day.