Wow you know a doctrine is bad when even the old covenant sounds better than what it teaches. Here's to you, you lousy "sin breaks fellowship with God" doctrine.
Our God, the great, the mighty, and the awesome God, You are a gracious and compassionate God who keeps loving kindness and in Your great compassion does not forsake. (Nehemiah 9:31-32)
You, O Lord, will not withhold Your compassion from me; Your loving kindness and Your truth will continually preserve me. (Psalm 40:11)
If I should say, “My foot has slipped,” Your loving kindness, O Lord, will hold me up. (Psalm 94:18)
This last one is foretelling the covenant of Christ's blood, the one you live under. This is a hearty taste of how good you have it with God.
"With everlasting loving kindness I will have compassion on you” says the Lord your Redeemer. "I have sworn that I will not be angry with you nor will I reject you. For the mountains may be removed and the hills may shake, But My loving kindness will not be removed from you, And My covenant of peace will not be shaken,” says the Lord who has compassion on you. (Isaiah 54:8-10)
Think sin does break your fellowship with God? Then read my post Inseparable & Unforsakable, Fellowship Never Broken