Fisrt of all, welcom aborad buddy. In here you can find many people who can help you and places to share your situations and experiences that might help other people.

Well, i've never had a relationship, BUT... I can say you actually love Lilly and are trying to help her, and that's a good sign. Now, she already acknowledges her insecurities, so the best for both of you is encourage her to talk about them with smoeone who can help her (Therapist or something). As i understand, you don't actually give her any good reason to make her think you would cheat her.

Good luck and welcome!