In chapter 8 of Adam McHugh's book, Introverts in the Church, he lists some practical steps for helping introverts thrive in evangelism (pp. 183-184). They are...
1. Narrow your focus. Rather than scattering yourself, focus on building relationships with one or two people with whom you feel comfortable and who have displayed spiritual curiosity. Work in small steps.
2. Ask Questions. Open-ended questions cultivate relationships, and at the right time, have the power to unlock people's deepest desires. Jesus would often respond to people's questions with another question, which forced them to account for their motivations and assumptions.
3. Ask for time. Introverts don't do as well when put on the spot. Instead of stuttering out an unprepared answer to a seeker's question, ask for time to think about it. "I don't know but I can find out" is a perfectly legitimate answer.
4. Don't accept the premise. West Wing chief of staff Leo McGarry taught his press secretary never to "accept the premise of the question." Sometimes unbelievers will come at us with a hostile posture and ask us an accusatory question, like 'How could you possibly believe in a God who would condemn people to hell?" If we accept the premise that God is a villain, we are put in a defensive stance and we let the other person control the conversation. Instead, rephrase the question. For example, you could respond, "Perhaps the real question is how could humans rebel against a God who created such a beautiful world?"
5. Find a comfortable environment. Consider participating in an Alpha class or in a Bible study for spiritual seekers. You could even start by joining an online forum for people with questions about God or creating a website designed for these sorts of discussions. Avoid debating situations that have no real goal other than disproving the opinions of others or just debating for debating's sake.
6. Know your role. You don't have to play every conceivable role in the great evangelistic task of the church. Focus on your strengths and what contributions you have to make to others. The best evangelism is done in community. We are part of a body of believers with different gifts and strengths - some have gifts of hospitality, others of service, others of teaching. A healthy community applies their gifts not only for the up-building of each other but also for the sake of those on the outside.
(This is not in its entirety, but gives you the basic idea).