Yes, there are many non - explicit Percabeth fanfictions that capture their intense emotions. For example, some focus on their arguments and subsequent reconciliations in a more platonic or romantic way without any sexual content. You can find them on fanfiction platforms by using keywords like 'Percabeth conflict and resolution' or 'Percabeth emotional moments'.
Sure. There are quite a few. A lot of fanfiction writers like to explore the dynamic between Percy and Annabeth in different situations. Some might write about them having a big fight over a mission or a decision, and then showing how they work through it and strengthen their relationship without any inappropriate content. You could try looking on popular fanfiction websites and filtering for Percabeth stories with a focus on their relationship drama.