I agree with 

Soredemo, Ikite Yuku - both the main male and female leads are adorably introverted who grow in their meeting each other
Date - similar concept, though this drama is more light-hearted
Love Shuffle - one of the males is a former bully victim who grew up trying to buy relationships, and one of the females is strange, depressed, and suicidal

Also, I'd add

Saikou no Rikon - main male lead is a very high-strung loner type whose personality leads to a failed marriage
 
These are some of my absolute favorite dramas!

Nobuta wo Produce
Date - Koi to wa donna mono kashira

Love Shuffle isn't like kokoro connect; It's actually a funny, inspiring and sweet story. You can ask yourself which woman you resemble the most and which man is more your type (and the reverse, for men) Sometimes it's serious, sometimes funny, sometimes meaningful, sometimes it doesn't make sense... It's a unique story with great characters
The jdrama Date - Koi to wa Donna Mono Kashira, has both a socially awkward heroine and hero!

Others jdramas with socially awkward (misunderstood) heroines include:
Keizoku 2
Hard Nut!
Nobuta Wo Produce

Not many kdramas that I know of with socially awkward female leads:
Oh My Ghost
Merry Mary

:)
Nobuta wo Produce
Yamato Nadeshiko Shichi Henge