Puppy love dramas

• Yes, Her Majesty (I'd say this is the closest to what you're looking for - ML is madly in love with the FL from childhood and does everything possible to make her happy even when it means giving up his position of power to her.)

Also maybe...

• Colourful Bone (ML is very traumatized, FL is his savior, and he's very puppy-like with her - following her, devoting himself to her, etc.)
• Happiness 
• Exclusive Fairytale 
• City of Streamer 
• Koi Desu: Yankee-kun to Hakujou Garu 
• Strong Woman Do Bong Soon 

Assistant of Superstar - This is the drama that first came to mind. The ML is so smitten with her it's adorable. He falls for her in the first couple of episodes and tries to get her to fall for him, all while being jealous, but in a cute way. He's not only so in love with her, he helps her get over her fear so she can finally become a teacher. It's literally so adorable.

Moonlight- This drama and the one listed above are very similar.  If you like one you might like the other. Very similar vibes. Both have a "famous" ML who falls for the FL very quickly and is smitten with her through the whole drama. The ML in this one is just so funny cuz he gets jealous so easy. And it's not a creepy/angry jealous, it's a cute, "no I like her!" jealous. I definitely recommend. 

Exclusive Fairytale- +1.  ML is such a green flag here, you've got to watch it. He falls first and of course try to get her to fall for him. Just overall a cute watch.

Forever Love - Another green flag drama. This drama the leads start to fall for each other in high school, but go their separate paths. The ML made it his goal to get into the same college as our FL, so he can win her over again. Honestly such a cute watch. The ML is so in love with her and does a lot of adorable things to win her affection. And the side couples are cute too. So it's a bonus.

What's Wrong With Secretary Kim?- This is a well known drama, but thought I'd mention it anyways. The ML is obsessed with our FL. Once he learns that she is going to resign, he tries to get her to fall in love with him so she can stay. He does everything to get her to stay, he even proposes multiple times. lol. But while he's doing this, he learns that he really does have feelings for the FL and realizes that he wants her, not just as his secretary. 

Dear Little Mermaid- The ML has loved the FL since their high school days. But she leaves because of an illness he did not know about. When he meets her again years later (while she's running from the hospital.. lol)  he decides that he's not going to let her go this time. He has plans to make her fall for him again. It's pretty funny what he does to get her to like him. And the way he teases her, is not only funny but adorable. Once the ML finds out about her illness, he takes it upon himself to take care of her and love her even more. It's so cute. BTW.. this drama isn't sad or anything like that, it's quite the opposite.

King the Land - ML is very in love with FL. He falls for her within the first couple of episodes, and from there, he makes it his mission to have her fall in love with him. It's hilarious, and adorable. The ML will not disappoint if you're looking for relentless pursuit.. 

Additional Dramas you might want to try out that are similar to what you're looking for..

Professional SIngle   Fall in Love With a Scientist  My Little Happiness  Unforgettable Love  Skate into Love

• When the Camellia blooms
• Secret Garden

This is my favorite kind of male lead so I made a list. ? Puppy Male Leads (108 shows)

• + 1 My Little Happiness - he's super in love with her from the start (it's a fateful childhood crush sorta thing) and he is constantly working to help her, even behind the scenes. 
• My Fated Boy - he's loved her for a long time and tries everything to stay close to her. 
• Weightlifting Fairy - one of the most adorable puppy MLs of all time imo. You can practically see the hearts in his eyes when he looks at her. 
• Something in the Rain - he is SO in love with her hahaha
• The King's Affection - a super fun role reversal where the FL is the cold and aloof one and the ML is the adorable clumsy mess who's trying to keep up with her while also crushing hard on her
• Who Rules the World - he's not the classic puppy dog ML but he is sweet and gentle toward the FL and he always tries to stay close to her.
• Shopping King Louie - adorable, sweet, clingy ML with a strong and competent FL
• Put Your Head on My Shoulder - omg. Love this ML. He is sweet and gentle and awkward, and he comes up with all kinds of silly reasons to convince the FL to spend more time with him
• Meow the Secret Boy - soft, sweet, clingy ML
• Meeting You - similar to Put Your Head on My Shoulder, the ML is introverted and a little awkward but opens up in a cute way when he falls for the more loud and extroverted FL

I have a list for MLs who are constantly seeking out the FL's company and touch, maybe you'll find some more good candidates there.