What do I do????

Ok so my boyfriend lives with his ex because he has no where else to go.....I know it already seems sketchy.....but it gets even worse. We were sitting on the couch and his "ex" texted him and she said babe....so I was like why is she calling you babe, and he just said she is crazy and still likes me that's all. But I told him that I don't like that, and he told me I have nothing to worry about. But then the next day he showed me messages of him telling her to back off and that he loves someone else. Should I believe that he is being faithful or should I leave him?????