Is da'wah an obligatory to all Muslims? What if I am the type of person who dislikes approaching people to give da'wah about Islam, while I myself have an inner turmoil about Islam I need to calm down and fix? It's the religious people in my country whom insists that it is obligatory for Muslims to meet others face-to-face and spread Islam with tongue, but I find it uncomfortable to do it. Am I sinning? I'm still learning Islam, but not from their source.
The type of dawah (proselytization) you mention is not obligatory and is in fact more likely to do harm than good. Those who believe in knocking on people’s doors and having really awkward interactions with them where they try to pass on Islam to them have a very demeaning attitude toward fellow humans. They think of other people as instruments for furthering their own agenda. That is neither a humane nor Islamic attitude to have.
Religion is a very personal matter and most of the time it is rude and unproductive to bring into a conversation. The way you call people to Islam is first through being a good example yourself, and secondly, you can do it through speech with those you are on intimate terms with, or those who have come to you asking you to speak to them about religion.
As for inconveniencing people to try to sell them religion, the Quran never tells us to do that. The Prophet Muhammad 
was merely calling his own extended family to Islam rather than knocking on strangers’ doors. And Moses, when he was calling the Pharaoh to become a believer, was actually talking to his own family, since he was raised in the Pharaoh’s family.
I am not against someone going to a park where people usually hold speeches and trying to speak to people about Islam. This does not inconvenience anyone and people are free to listen or leave. A person who has the motivation and ability to do that can do it. But this is not for most people.
For the average Muslim, “calling people to Islam” means to be a good example. When it comes to people you are very close to, such as a close friend, you can try to encourage them to be better Muslims, or if they are not Muslim, to inform them about Islam and why you believe in it. But this should be done when it comes up naturally in conversation. One of the main examples of dawah in the Quran is in that of the two friends talking in Surat al-Kahf:
His friend said to him, as he conversed with him, “Are you being ungrateful to Him who created you from dust, then from a sperm-drop, then evolved you into a man? (18:37)
The Quran makes it clear that they were already talking, and that this was a natural and relevant addition to the conversation; he was not just randomly bringing up religion. The irreligious friend had said this before:
And he entered his garden, wronging himself. He said, “I do not think this will ever perish.”
“And I do not think the Hour is coming. And even if I am returned to my Lord, I will find something better than this in return.” (18:35-36)
The friend brought up religion, and the religious friend used that as an opportunity to try to tell him to come back to fearing God.
And if a Muslim has a friend who is a Christian and who is not interested in speaking about religion, then harassing them with religious talk should be the last thing we do. They are humans just like us; we should never treat them like objects, trying to change them into something else without their own willingness and participation.
A good rule of thumb, therefore, is that if dawah feels rude, uncivilized and awkward, then it is wrong. Only when it comes up naturally and when the other person is interested in listening is it right to do it. Dawah in public places, where it is not a nuisance and when people are completely free to listen to it or to ignore it, is also fine.