We talked last night and I told him I was open to him bringing someone else into his life. He swore he was not having a sexual relationship with anyone, and that he did not want to open the relationship because he was worried about him being jealous of my other partners. I couldn't bring myself to admit that I had seen the text messages, but I do feel like we made progress.
Thanks to everyone for your advice. I think I'm going to watch for other signs and confront him when it seems logical. Afterall, it's not like I want anything to change other than the hiding.