Its me again, and thanks for the recommendations to my other threads. About the topic, I do prefer something that does not feel like they're acting, like they are naturally awkward and not "acting" awkward. Should have good story, something that seems addictive. nothing average (like it seems that I could've watched something better)

I only know:

- Mars (currently watching)

- The Great passage

- Tomdachi no hanashi

- Kimi ni Todoke (did not like the live-action)

I like this theme but it’s hard to actually find Asian dramas and movies that fits it.

The first thing that came to my mind was a movie Sukitte Ii nayo, although I don’t think that it was great or anything special it was just good but fits the theme, It seemed a little bit similar to Kimi ni Todoke.

I recommend a drama Cat Street – this one I really liked and thought it was good but too short.

There was also one girl in Love Shuffle but she was one of the many characters so she wasn’t like the center of a drama.

Oh almost forgot, There is also korean drama Flower Boy Next Door . I thought it was very good drama.
1. Yeah, I've read there's nothings special about the live action of sukitte. I've read the manga, watched the anime, and watched the trailer of it haha
2. Cat Street- thanks, but I might read the manga instead due to first review on its MDL page
3. Love Shuffle- Is it like kokoro connect (anime, but haven't watched it)? Thanks, but I kinda feel I won't get addicted to it
4. Flower Boy Next Door- I actually tried watching it awhile ago, it was adorable. I also like the female actress hehe. But, I get really turned off the 'surgeries-actors and actresses' I AM SO SORRY but I think a lot of koreans are beautiful and do not need to undergo the knife. But I think it will be a show that I woukd not regret on watching
I can't think of any aside from Heart to Heart. The fact that she can't interact with people is the main plot, essentially. It's super cute. :3
I second Flower Boy Next Door, even though I never finished it (I'm not a great fan of the male lead)

I really enjoyed a taiwanese drama called ToGetHer, although it wasn't to everybody's taste, mainly because many thought the female lead had no backbone. To me she came across as a strong character in an understated way, and yet at the same time she could be vulnerable and emotional. She's a real introvert and only really talks to three people throughout the drama, but those she talked to were the ones she truly cared about. The male lead, Mars, is a disgraced star desperate for recognition and appreciation (typical opposites attract setup). I ended up liking him a lot though, because he cares so much about her and helps her as much as he can. His behaviour with her was very different than with other people: unlike his usually obnoxious attention-seeking star attitude, with her he was always gentle, quiet, mindful, insecure about himself yet determined to help her. And at the same time, she's the one that helps him the most. 
I recommend Adolescence Medley.
The guy feels somehow awkward (me and my friend both agree on that).
And I think it has a good story and not average, in fact it's one of my favorite drama. It's short and super cute !
Agree w/ Heart to Heart. It's fantastic.
i'm not very sure about what kind of awkwardness you are looking for, so i'll write some dramas that come to my mind:
- the main character of Incomplete life (Misaeng) seems very awkward (plus the drama has very high rating, so i hope its good since i havent seen it yet but i plan to do so soon)
- the main girl of High school king of savvy was.. awkward.. (BUT! i hated that drama so. much. Hovewer, there are 2 kinds of ppl. some hate this drama and some love it so...)
- some characters in It's okay it's love could be considered bad at socializing but i'm not sure if you are looking for this "type" of problem because this drama is dealing with some mental issues
- maybe also Nobuta wo produce. but i dont remember much about it
Soredemo Ikite Yuku. The heroine and hero are socially stunted from a childhood tragedy. 
I Need Romance 3: heroine has been hurt emotionally and lacks empathy
Master's Sun: because of the whole ghost situation, she doesn't get out much and seems strange to other people
Yamato Nadeshito Shichi Henge: a bit silly, but the heroine is into horror and loves to be left alone
Your description is perfectly suited to the drama 'High School King of Savvy'. Lee Ha Na who plays the female lead is so socially awkward, it was painful to watch at time- like the time when she was in the car with her boss and was sweating so bad and he saw it hahahha urgh 
Yamato Nadeshiko Shichi Henge- the first thing that came to mind and also heart to heart.
 Exo next door, has kind of thing as well if you want something quick and cute to watch.
I just finished The Rose & the heroine is quite socially awkward in it. Maybe you'll find it interesting. Does cover some taboo subjects though, so depends if you mind that or not.

I know you said Cat Street, Love Shuffle & Flower Boy Next Door didn't really appeal but all 3 are really good & addictive. They definitely would suit your criteria.