Although I am a big fan of action-adventure games with good stories, twin-stick shooting is my least favorite mechanic in video games, as well as my most least experienced genre. One twin-stick game I did enjoy was Nuclear Throne.
I generally enjoy games that tell stories in different ways. There is no particular game in my gaming history that is close to this, but one game that I liked that told a story in a unique way was Her Story.
I am very drawn to games with new concepts that try to bring a different experience to the player through its execution. Journey is a good example of a game that did that successfully that I respect.