First, think about the character's backstory. If they're a dragon shapeshifter, what led them to be this way? Were they born as one or was it the result of a curse or a magical experiment? This backstory will shape their personality. In terms of appearance, when in human form, they might have small details that give away their true nature, like a birthmark in the shape of a dragon scale. Their behavior should also be consistent. If they're used to the freedom of flying as a dragon, in human form they might be restless or have a love for high places. Also, their relationships with other characters, whether they be other shapeshifters or normal humans, should be complex and realistic.
To create a believable character in dragon shapeshifter fanfiction, start with their dragon form. Decide on its color, size, and special abilities. For example, a black - scaled dragon might have the ability to breathe shadowy fire. Then, when they shift into human form, there should be some connection. Maybe their eyes remain a dragon - like color. Also, give them a personality. Are they fierce like a dragon even in human form? Or are they more gentle, trying to hide their inner dragon.