I really believe she has latched onto the idea of saving Abraham because it is something concrete. Just like Ichabod was off chasing rumors of demons, she was off trying to find something to distract her as well. Abraham is the only person in the entire world, other than Ichabod, that Katrina has any connection to. She doesn't even know if Henry is alive right now. She knows she can go to Abraham at any time. She knows she can talk to Abraham. What does she know about Ichabod? Nothing. He's been maintaining a separation from her. As screwed up as it is, Abraham is more real to her than Ichabod. That isn't about love or care, it's desperation for something solid in her life, a life that has been riddled with transient and fleeting interactions for entirely too long.