Fear, worry, and anxiety are just feelings - just like anger, guilt, and annoyance. They are, to be sure, a natural part of our humanity. But they are just emotions, and they don't have to govern us and become sin.
God created us as emotive beings. Our emotions are part of who we are. Each of our feelings is part of our divine design and has its place. But, due to the fall, we must keep them in their place.
To be sure, our problems are not our emotions. Our problem is letting ourselves be ruled by our emotions. When we allow our feelings to dominate us and dicatate our behavior (and not the Word of God), that is sin.
Being annoyed with the faucet dripping is not necessarily wrong. That initial annoyance may be said to be a natural feeling. Bashing the sink in with a sledge hammer or yelling at your husband for not fixing it would be sinful. You've just let your emotion drive and control you.
The same applies to fear and worry. Fear of your child being abducted is not necessarily a bad thing. Hovering over them every minute of their lives and not allowing them to take risks or carry out normal life activities is sinful. That fear has come to dominate your behavior. Parenting is no longer about raising a responsible and productive servant whose life is in the hands of God; it is about my control and squelching life.
John Piper once said that "God designed your emotions to be gauges, not guides. They’re meant to report to you, not dictate you."
Those are beautiful words. When it comes to fear, it is a good thing. Fear is an emotion that reports to us: it tells us something is not right. It reports the fact that there is danger; the potential for harm.
While worry or fear may be a God given guage, it is not a guide. God's word is to be our guide. Thus, we must have the Word of God rule over our emotions and we must tell them how far they are to go. Moreover, we must will what is right in the midst of them.
Did David fear Goliath? Certainly not like the rest of the Israelite soldiers who cowered in their foxholes. These men (if you can call them that!) were guided by fear. The fear had become their boss. They would not serve God, but served the fear.
David may have been scared. Fear may have rightly said, "There's a human tank out there." But even if he did have some level of concern/fear (we are not told exactly), we do know that he courageously ran towards the big oaf and accomplished the Lord's will. Even before going to the battle feild he entrusted himself to God with his words.
When the Bible tells us to "Fear not" or "Do not worry," it is telling us not to let these emotions drive us or divert us from our God given service. When we are annoyed, we actively work to suppress the desire to act on that annoyance. Instead of lashing out, we try to tenderly deal with the source of our annoyance.
When we are fearful, we do the same. We suppress the desire to run, stew, cower, or fret. Instead, we think of God, thank Him for this circumstance to prove Himself to be mighty, call upon His streghth and protection, and actively will to do as He would have us.
Fear is good. It is a guage. We can thank God for the emotion of self preservation (or better yet, the emotion which indicates our need for God's preservation). But it is not a guide. It is just a feeling which, along with all other emotions, must submit to God and be kept in its place.