Some that I have seen and loved are Healer, Strong Woman Do Bong So, Shopping King Louis, Legend of the Blue Sea, Goblin, Moon Lovers: SHR, etc...
All of the dramas I have seen are here: MyDramaList It's okay if you recommend me dramas that are already on my to-watch list.

Also it's a bonus if the male lead falls in love first, but the main thing I'm looking for is that the male lead is clingy, but not like a super possessive clingy like in Heirs where he treats the female like an object. More like in Healer when he does this kind of stuff:
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OMG so cute... I can't
I was just going to say that you wrote a description of the male lead in Strong Woman DBS :D

I would add One More Happy Ending
Noble, My Love (maybe too clingy and controlling)
Fight My Way (definitely super jealous)
@Linus Thanks for the recommendations... Fight My Way had been at the top of my want to watch list, but I've been hesitating because it seemed like the male lead was mostly disinterested in the female lead.
Now I will definitely be watching it. Thanks!! :D
Actually, they have a pretty awesome friendship where he regularly gets jealous. I think you would like the interactions between them based on your list ;) Noble, My Love is about a very clingy, jealous and controlling guy. It's a little more severe version of Strong Woman ;)
my secret romance (same male lead from noble,my love) he fell in love first , and was clingy AF :) .. bonus: so many hot kisses ;)
Boss and Me. Male Lead falls first, is very jealous when her attention is not on him, and connives in all kinds of sneaky ways to make her interact with him. 
Lots of skinship too.

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Faith

Cruel Romance

Prince Of Wolf
Good ol' Boys Over Flowers
Thanks for all of the recommendations!! I can't wait to try some of these out especially Faith and Boss and Me. (The first because of Lee Min Ho ;) )