Boyfriend texted ex GF

My boyfriend dated this girl (we'll call her Sara) for about 6 months. They had a bad relationship and ended up breaking things off. About 2 months later, we started dating. About 2 months into the relationship, we had a bad breakup and he got super drunk. So drunk that he texted Sara saying he loved her and only dated me to get over her. He was also texting this girl from Tinder flirtatiously at the same time. The next day, he had to go to the hospital because he took the breakup so badly and got really drunk. But later on that day, when he was sobering up, he still texted those two girls. He even planned to meet up with his ex because she said she really needed to talk to him about how their relationship ended. She ended up cancelling on him. Well, we got back together and I had heard a bunch of rumors from his ex about how he had been texting her. I was so hurt, but I tried my best to hear out his side and believe him. He said he never texted her. I foolishly believed him, but later looked at his messages to find out the truth. He told me he lied because he didn't want to loose me again. I didn't believe him, we split up, and eventually got back together again. We had a serious talk about boundaries, honesty, the whole thing. Since then, we've been together another 5 months and we've overall been happy. He doesn't text other girls, especially not his ex, and we've been good. However, I can't shake the feeling that something like this might happen again. He wants to marry me, and Im very hesitant, because of what happened in December, and how he lied to me about it. Any advice on what I should do?