Should I break up with my live-in girlfriend?

I'm a male in my 20s. My girlfriend is in her late 30s. She's great. She's funny and smart, she has a big heart, and we have an excellent sex life. She recently moved in with me partially because she wanted to and partially because she had no place to go. We fight a lot. It's mostly my fault, I must admit. She doesn't like my insecurity and lack of trust I have for her. I have trust issues. Also, I can't fathom why a woman like her is with me, so I'm always dreading when a better dude will come along. I don't think she's happy. She's very submissive and she loves me very much, but also the fact that she has nowhere to go must be influencing her decision to stay. I love her so much, but my jealousy is not likely to diminish. I never believed in the whole "If you love them, let them go," but I do now. I really want her to be happy. Should I end it with her? She has no place to go so I feel like I can't break up with her. I'm literally trapped.