The story of Captain Carter in the comic often involves her facing unique challenges and using her skills to overcome them. There's a lot of focus on her character development and the alternate universe's impact on her.
One possible plot could be that they meet in an alternate universe where timelines have merged. Captain Carter and Captain America initially have a bit of a misunderstanding as they come from different versions of heroics. But then they realize they share the same values of justice and freedom. They team up to fight a common enemy that is threatening both their worlds, like a powerful time - traveling villain who wants to rewrite history to suit his evil plans.
Red Skull could be the villain. Since he has a long history with Captain America, in a fanfiction where Captain Carter is involved, he could be resurrected or have found a way to cross over from another dimension. He would see Captain Carter as an interesting new threat and Captain America as an old foe to be finally defeated.
The theme of sacrifice is also quite common. Steve sacrificed his life (or so he thought) by crashing the plane. Peggy sacrificed her personal life to a great extent for the SSR and the fight against Hydra. Fanfics will often put them in new situations where they have to make more sacrifices. And then there's the theme of time. The fact that Steve was frozen for decades while Peggy aged is a rich area for exploration in fanfics. They deal with how time has changed them and yet how some things between them remain the same.
Writing their relationship well means exploring the power dynamics. Captain America is a symbol, and Sharon is a strong agent in her own right. Show how she respects his position but also challenges him when necessary. Their relationship could be about mutual growth. For example, Steve could have taught her about leadership, and she could have taught him about modern society. Additionally, add some conflict. Maybe they have different views on how to handle a particular threat, which tests their relationship but also deepens it in the long run.
A possible storyline is Sharon Carter getting involved in a time - travel adventure similar to what Steve Rogers had. She might end up in a different era and has to use her skills and wits to survive and find her way back. Along the way, she could discover new things about Captain America's past that were previously unknown. Or perhaps a story where she teams up with other superheroes to take on a new threat, but she's constantly haunted by her relationship with Captain America and how it affects her actions in the team.
Another storyline might involve Sharon Carter getting kidnapped and Captain America having to rescue her. This would not only show his heroism but also his deep feelings for her. Along the way, he might have to fight his way through hordes of bad guys and overcome various obstacles.