A key Bible passage some Christians use for teaching that believers have at least SOME responsibility to bear witness to the truth in Christ is Ezekiel, chapter 3:
16 At the end of seven days the word of the Lord came to me: 17 “Son of man, I have appointed you a watchman 23 for the house of Israel. [The literal role of a watchman is described in 2 Sam 18:24; 2 Kgs 9:17].Whenever you hear a word from my mouth, you must give them a warning from me. 3:18 When I say to the wicked, “You will certainly die,” 24 and you do not warn him – you do not speak out to warn the wicked to turn from his wicked deed and wicked lifestyle so that he may live – that wicked person will die for his iniquity, 25 but I will hold you accountable for his death. 26 3:19 But as for you, if you warn the wicked and he does not turn from his wicked deed and from his wicked lifestyle, he will die for his iniquity but you will have saved your own life.
3:20 “When a righteous person turns from his righteousness and commits iniquity, and I set an obstacle 28 before him, he will die. If you have not warned him, he will die for his sin. The righteous deeds he performed will not be considered, but I will hold you accountable for his death. 3:21 However, if you warn the righteous person not to sin, and he 29 does not sin, he will certainly live because he was warned, and you will have saved your own life.”
If we miss the metaphorical import of this passage, we may be in danger of becoming neurotic in our attitude toward unbelievers and BELIEVERS alike ("This viewpoint has periodically reduced me to a nervous wreck"--your words).
As a watchman for Israel, Ezekiel was accountable to warn both the unrighteous and the righteous--unbelievers and believers alike. For Ezekiel to shirk his duty as God's mouthpiece (which is what a prophet is, in essence) would be to fall down on the job to which God called him.
Ezekiel did not fall down on the job, however. He was faithful in warning the unrighteous and righteous of the punishment which awaited them if they never initially repented of their sin or if they had repented at one time but "backslid" into their old way of life.
The application of this passage to Christians today is pretty clear: We, too, are watchmen and watchwomen. We have been saved from wrath through Christ and his cross-death. Out of gratitude for God's gracious and loving salvation, we are to be Christ's witnesses in the world (a witness is analogous to being a watchman who warns people of approaching danger).
Jesus: ". . . but you will receive power when the Holy Spirit has come upon you; and you shall be My witnesses both in Jerusalem, and in all Judea and Samaria, and even to the remotest part of the earth" (Acts 1:8 NAS ).
Does this mean that we as Christ's witnesses should go around in a continual state of anxiety if we fail to witness to every person with whom we come in contact? Of course not.
"Have no anxiety about anything, but in everything with prayer and supplication with thanksgiving, let your requests be made known to God. And the peace of God which is beyond human understanding will be like a garrison of soldiers, keeping your heart and mind in Christ Jesus" (Philippians 4:6-7, rhetorician's version).
We Christians are witnesses, to be sure. We are also ambassadors for Christ (see 2 Corinthians 5:20). When we think of witnesses today, we automatically think of witnesses in a court of law. Witnesses are to "tell the truth, the whole truth, and nothing but the truth, so help them God." Outside the context of a court of law, however, there is another way of witnessing to the truth, and that is by living a life which is worthy of Christ, being light and salt in a corrupt and dark world, shining forth as lights (see Philippians 2:15).
Going back to Ezekiel, we see that Zeke was responsible to warn both unbelievers and believers alike. The same is true of us. The way to go about doing so is by witnessing by both life and lip, but leading with our life. A saying I grew up with, which I learned probably from a Roy Rogers(!) song (Rogers was a television- and movie star in Westerns, mostly, who was also an outspoken Christian), and the song was probably based on the words of Ralph Waldo Emerson:
"What you are speaks so loud, that the world can't hear what you say!"
Does this mean we don't have to open our mouths and bear witness to the truth as we've found it in Jesus? No, but it's not a matter of either/or, but both/and. We are to live exemplary lives--even if it means suffering for righteousness' sake, and we are also to open our mouths and tell people of what we have found to be true. The truth, of course, is contained in the Bible. John 3:16 is as good a place to start as any, I suppose.
Too often, some Christians are so eager to "share" their faith that they lose all perspective and make a mess of being Christ's witnesses. They witness out of guilt. They witness without love, tact, and gentleness. They witness without sensitivity to the Spirit and to the needs, questions, and hang-ups of the people to whom they witness. They mean well, but they are poor ambassadors who perhaps do more harm than good! They need to take to heart 1 Peter 3:14-17:
For who is going to harm you if you are devoted to what is good? 14 But in fact, if you happen to suffer for doing what is right, you are blessed. But do not be terrified of them or be shaken. 15 But set Christ apart as Lord in your hearts and always be ready to give an answer to anyone who asks about the hope you possess. 16 Yet do it with courtesy and respect, keeping a good conscience, so that those who slander your good conduct in Christ may be put to shame when they accuse you. 17 For it is better to suffer for doing good, if God wills it, than for doing evil [my italics].
Notice how Peter tells us to be "ready to give an answer." That means we are to wait until people question us, and not vice versa. When they ask us about why we seem to be at peace, why we live by a higher standard than many people, why we seem to care about people, and so on, we then are free to "give an answer" to them. Giving an answer, by the way, means literally "to make a defense," which of course is the art of apologetics. That, however, is a subject for a different question!
In conclusion, I think we need to take a cue from our Lord's approach to people, in that very frequently he asked leading questions and comments--you know, the open-ended kind which need to be answered with more than just a yes or no (see John 3 and 4, in his interaction with Nicodemus and the Samaritan woman at Jacob's well). That approach, especially when it is preceded by prayerful dependence on the Lord to give us the right words and attitudes, goes a long way toward fulfilling our obligation as "watchmen" for God.