I've been both a victim and a witness of many situations where coming from a broken family has lead to a fear of abandonment. Let me tell you this: be cautious of how close you get. There's a certain point where a close but healthy relationship crosses over to a codependent relationship. Do not, DO NOT let yourself become too clingy. Be wary of the desire to be involved in her entire life; it will ruin everything.
All that's good advice for what NOT to do, but you probably also want advice on what TO do.
>>5 has the right idea. I would recommend just telling her about the problem and what you're going through, as well as your fears of what it's doing. If she understands that, she'll probably be willing to even help you. Tell her to let you know when she feels you're projecting your fear on her, that way you can control it better.