Blink

The killer is after people who received donated organs from a girl he was obsessed with. The victims were found with their wrists slit because this made the organs useless. There was an error in Emma Brody's (Madeleine Stowe's) hospital admissions form, which led the killer to her neighbor by mistake. Detective John Hallstrom (Aidan Quinn) saves the day, as Emma finally adjusts to her new eyes. Emma and John fall in love, of course.
Thanks, Sharla D. and Tuxxer!