conversation

A: What's eating you?
B: Because of the incident.... <---This doesn't make much sense in English, we would say, "Thinking about what happened..." Or "Last night..." We would reference the time in which the incident ocurred, we wouldn't say, "Because of the incident."

A: What? What are you talking?

B: Actually, I was drunk off my head at the bar when I drank with my co-workers. As you know, I really love drinking, so I just drank without any thought. Therefore I blacked out. I think it was really shameful behavior.  

This is gramatically PERFECT. But, it's unnatural. Try, "Well, I went to the bar with my coworkers and I got drunk out of my mind. But you know, I love to drink, so I just kept drinking without really thinking about it. I ended up blacking out. Shameful behavior, I know." 

That sounds way more natural and conversational. But of course, I am Southern, from the deep south of the US, so you may get other variations but the general idea is to relax. We use buffers a lot in English to make our speech less formal when in conversation, and this topic isn't formal.

A: Why? Enjoying the moment is important. Don't be ashamed of this your behavior.

B: But.... Drinking too much, I threw up at my superior.

"But...I was so drunk I threw up on my supervisor."

In American English, we don't call someone our superior unless we're talking about the military. We say Manager or Supervisor, and we don't really use Elder unless we're talking about Senior Citizens. 

A: What!! But you said you blacked out. How've you known it?

"How would have known?" OR "How do you even know?"

Both are correct but the second one is less formal and sounds a bit more relaxed.

B: Because One of my colleagues texted me in the morning. Oh How do I deal with? I think I can't hold my head high at work.

"Oh, what should I do..? I don't think I can walk in with my head held high at work anymore..."

We wouldn't really say that but I get what you're going for. Hmmm, try something like "I'll have to tuck my tail between my legs at work tomorrow." That has the feeling you're going for.

conversation

A: What's eating you?
B: A certain incident....
A: What? What are you talking about?
B: Actually, I was drunk off my head at the bar when I drank with my co-workers. As you know, I really love drinking, so I just drank without any thought. Therefore I blacked out. I think it was really shameful behaviour.
A: Why? Enjoying the moment is important. Don't be ashamed of this behavior.
B: But.... After drinking too much, I threw up on my superior.
A: What!! But you said you blacked out. How do you know that?
B: Because one of my colleagues texted me in the morning. Oh, how do I deal with this? I don't think I can hold my head high at work anymore.