This is a love story, not a horror story.
She's relaxed and excited at the same time, enjoying her budding romance, not waiting for the trap to spring and his teeth to find her throat and her life, her blood, every part of her to drain away until she's a withered husk of herself. (Her life revolves around him by now, anyway, so even if that does happen she might not notice the difference.) She's in love with him, not afraid of him; intrigued, not repulsed. If she stares, it's because he's gorgeous (and what else? What else is there to him that attracts her?) not because her animal instincts are telling her to keep an eye on a possible threat.
How could he be a threat? It's a love story, after all.
She needs him to stay in her life, because she loves him. (A psychologist might use words like fixation, dependency, obsession.) He loves her back and wants her to be with him forever (How does she know that? How has he shown her his heart, besides not killing her already? What proof does she have of his feelings, of anything he's told her, besides his word?) and she wants that too. (Why? What does she think is so compatible between them that it can last even for a century, never mind for ever?)
This is a love story, so there has to be a happy ending … except immortality means there is no ending, so it's more like a happy eternity.
He's beautiful (the way that a tiger or a snake is beautiful) and protective (possessive) and everything short of actual royalty that a Prince Charming can be (What does she even know about him, his personality, his interests, really know?) so of course it must be a love story and that means it can't be anything else. She wants to be like him so that 'happily ever after' really does mean 'ever after,' not because that means she won't be prey anymore and can (maybe, probably, eventually, hopefully) stop fearing him. It's not like that, it can't be like that, because she doesn't fear him, really, not at all.
This is a love story, not a horror story (and if she tells herself this enough times maybe she'll start to believe it.)