Percy's anger could make his powers go a bit out of control. For example, if he's near water, the water might start churning violently as he gets mad while protecting Annabeth. He could also become more physical in his protection, standing in a very aggressive stance in front of her, almost like a wild animal guarding its mate.
When Percy is protecting Annabeth in fanfic and is angry, he might become more impulsive. Instead of thinking things through, he just acts on his rage. For instance, if someone insults Annabeth, he might immediately lunge at them without considering the possible consequences, because his anger at the insult to her is so intense that all he can think about is defending her honor.
He might start yelling. Percy is known for his passionate nature, so when angry on the Argo 2, he could raise his voice at those who he thinks are at fault.
Well, it could be shown through her getting jealous easily when other girls interact with Percy. For example, if a new girl in the story tries to talk to Percy, Annabeth might give the girl a cold stare or quickly insert herself into the conversation.
In the fanfiction, Percy's possessiveness might also be demonstrated by his need to be included in all of Annabeth's plans. He may feel left out or threatened if she starts making plans with other people without him. He wants to be a part of everything she does, almost like he owns a part of her time and attention.
Often, she becomes overly possessive. If she sees Percy talking to another girl, she'll quickly insert herself into the conversation. She might put her arm around Percy or give the other girl a look that clearly says 'he's mine'. This possessiveness is a clear sign of her jealousy in the Percy Jackson fanfiction world.
He might be a bit shy at first.
She might be initially very angry. Annabeth is intelligent and perceptive, so she would likely feel betrayed that Percy thought he could deceive her. For example, if Percy lies about a mission, she could find out the truth later and confront him immediately, demanding an explanation.
Percy might be a bit on edge. He would be constantly aware of his actions and words, not wanting to mess things up. He would probably listen attentively when Annabeth's family members speak and answer their questions as honestly as he can.
She might initially be shocked. Annabeth is a strong character, so she could start by trying to stand up to Percy verbally, asking him why he's being so mean.
He might be initially in shock. Percy has been through a lot with Annabeth, so being dumped would come as a huge blow. He could be speechless for a while, unable to process what has just happened.
He could become withdrawn. Percy is a sensitive person at heart. So if Annabeth's actions are really hurtful, like if she cheats on him in a fanfic (although it's a very sad scenario), he would likely isolate himself from her and their friends. He might go off on his own to think things through and deal with his pain privately.