Weightlifting Fairy Kim Bok Joo, Shopping King Louis and Oh My Venus. I love those. 

And i see many people write the same :P 
in this order :- 
Pasta
The Master's Sun
Oh My Venus
Goblin
Oh My Ghost
Miss Korea
Sweet Stranger and Me
Beautiful Gong Shim
High-end Crush
Coffee Prince
Shopping King Louis
Secret Garden
The Greatest Love
It's Okay, That's Love
Can You Hear My Heart
Me too, Flower!
When a Snail Falls in Love
Lucky Romance
Something About 1 Percent
Cain and Abel
Cunning Single Lady
King2Hearts
Uncontrollably Fond
Descendants of the Sun
Queen In Hyun's Man
LOVE 020. I am not even kidding, this is the sweetest drama in the world. If you've never watched  chinese drama, start with this.
Weightlifting Fairy Kim Bok Joo
Love O2O
Queen In Hyun's Man
Oh My Ghost
Lucky Romance
Healer
orange
strobe edje
heroin sikkaku
koizora
i love you
Golden bride (2007) is the only one I can think of that TOTALLY fits :) It's quite long but also pretty unique and have no unnecessary drama. The main couple becomes totally devoted and their relationship is amazing. Then there's Heart to heart that's pretty sweet, and Addicted heroin is pretty much totally monogamous and cute.
Splash Splash Love
+1 Shopping King Louis
Coffee Prince
Legend of the Blue Sea
Witch's Romance
Weightlifting Fairy Kim Bok Joo
I would give it 10/10 on sweetest and cutest 
Strong Woman Do Bong Soon 
Orange Days

Brother Beat - a family drama about 3 guys & their slacker mom, so 3 different, but sweet love lines and funny as heck too, as I recall! :D

Padam Padam - the couple take their sweet time getting together, but once they do, their love is pretty sweet.

It's Okay, That's Love

Someone Like You - so sweet, esp the beginning the episodes where the hero is blind & he hires the female lead to be his eyes. - Too

Too Late To Say I Love You - more passionate love than sweet, but there are some sweet, poignant moments scattered throughout and the leads are totally besotted with each other, so much so they'd defy anything & anyone just to be together. A bit of a roller coaster though, so if you're just looking for a sweet, easy going drama that will not in equal amounts enchant & infuriate you, this is not it.

It Started with a Kiss / They Kiss Again - they don't come sweeter than this. Japanese version is also nice, but I prefer this, the Taiwanese, version.

Hapimari: Happy Marriage? - Japanese attempt at a Cinderella type story ... not as good as Korean & Chinese/Taiwanese dramas in this genre because let's face it Japan doesn't do romance dramas well, at least not any more as far as I can see. There's hardly ever any touch connection between the characters, the love doesn't feel real & you're lucky if you get one kiss, much less a good kiss. Their '90s dramas were a lot better for the most part. But Hapimari at least makes an attempt. You even get one pretty decent kiss! haha

Already mentioned, and fully seconded by me because they fit the description of sweet really well:

Boss & Me
Golden Bride
Also forgot to mention MARS - sweet because of everything the female lead went through & how the guy helped her overcome.
Boss & Me
Something about 1%(2016)
Healer, Love O2O,
Beautiful Gongshim,
Noble,My Love
Weightlifting Fairy Kim Bokjoo
Heart To Heart
W, Goblin, Sweet Stranger and me, You Who Came from the Star
Probably not fully to your requirements, but I thought it to be very fun :) ATM: Er Rak Error