ArniA:
Kingdom zombie series (2 seasons + extra movie)
Ashes of Love (has supernatural "heavenly creatures", some are monsters, a lot of wuxia/xianxia series from china have these type of beings)
A Korean Odyssey (has "eternal" or "immortal" beings)
Goblin (well, isn't a goblin a monster?)
Maybe you need to specify what you classify as a "monster". Sweet Home is quite unique, it's the most monster-like series.
Thank you for the rec! I've seen all those except for Ashes of Love.
By monster action drama/movie, I mean like the main characters are being chased by or must find some kind of supernatural or sci-fi-esque monster. The monsters aren't "good" or sentient. They're generally mindless and very bad news such as Kingdom. That's a good one! Unfortunately, I've seen it ;-;
Another one I really liked is the drama "Search" with Krystal and Jang Dong-yoon.
I think the upcoming "Hellbound" falls right into what I'm looking for so I'm really excited for that!