I'm not @oliviasstephen, but I AM someone who is closer to the asexual side of the spectrum than anywhere else (you should come hang out; we have cake!), so here are my two cents.
Different asexual people have different ways of feeling attraction - some are comfortable with certain physical aspects, others aren't, some of them are a-romantic as well (that is, feel no interest in romantic love at all), some aren't; some are heteroromantic, some are homoromantic, or anything in between, around or beyond those two, etc., etc. - but it is DEFINITELY possible to fall in romantic love with someone without in anyway linking that to a sexual attraction.
You know that fluttery feeling of excitement over having a crush? That thing where it feels like your entire chest might be full of butterflies and sparkles and warmth, especially when you're close to the person you feel that way about? Asexuals have that too. They just don't link it to sex. Just as you can have sex with, and be sexually attracted to a person without being in love with them, it is entirely possible to be in love, romantically, with a person without wanting to have a sexual relationship with them.
But I think this thread is slowly sliding off topic; we've been discussing how to make a fictional romance work within the framework of a story. We're not here to argue about the validity of, or even the existence of, the love lives of people on various ends of the sexual identity spectrum.