Healer - the male lead doesn't "live like other people" to quote the drama.
Live Up to Your Name - time travel + goof ball = a good amount of weirdness.
It's Okay, That's Love - the fact that they're all dealing with their own weirdness is pretty much the premise.
Gu Family Book - half gumiho.
Pinnocchio - weird circumstances make the male lead weird. He gets a nice makeover later.
Moon Lovers: Scarlet Heart Ryeo - male lead is an outcast in his family because of his mother, circumstances of his upbringing, and the scar on his face. He's a bit dark and on edge as a result.
Kill Me, Heal Me - multiple personality disorder.
Goblin - title says it all.
You Who Came from the Stars - 400 year old alien is still learning how to interact with people.
Strong Woman Do Bong Soon - male lead and female lead are both weird in their own ways.
Introverted Boss - probably more what you were looking for. The boss has a certain level of social phobia / is extremely shy.