One great Edward fanfiction is 'Edward's Adventure Beyond the Bebop'. It shows Edward exploring a new planet filled with strange creatures. Another is 'Edward and the Lost Code' which has Edward solving a mysterious code that could save the Bebop crew. And 'Edward's Cybernetic Discovery' is also interesting as it delves into Edward's relationship with her own cybernetic enhancements.
I think 'Edward and the Enigmatic Signal' is a great one. It starts with Edward detecting a strange signal, and as she tries to figure out what it is, a whole mystery unfolds. There are twists and turns as she investigates further. The way it's written really makes you feel like you're right there with Edward.
One common theme is the contrast between Edward's vampiric nature and the cowboy lifestyle. It creates an interesting dynamic as the cowboy way of life is often associated with freedom, roughness, and a connection to the land, which is different from his refined vampire self.
You can try searching on fanfiction websites like Archive of Our Own (AO3). There are often a lot of Twilight - related fanfictions there, and by using the search terms 'cowboy Edward', you might find some interesting stories.
The true story of Cowboy Charlie Siringo is one of adventure and hardship on the Texas range. He faced many challenges but persevered with his cowboy spirit.