Leave or stay

My background: married first at age 21 - decades long marriage to a man with integrity but whose values evolved to be very different and in opposition to mine --> I no longer felt accepted --> I sought divorce. My ex and I now have a good relationship, occasionally talk on the phone about the kids, and I wish him the best, and have done my best to help our adult kids to maintain a loving relationship with him.) Eight years ago I met a man and we fell in love. He presented himself as someone who had screwed up a lot and made lots of mistakes in his past, but that he had "learned his lessons" and "was ready for a real and mature relationship" (he had been married ad divorced 4 times). He really fell in love with me, and over the years kept saying he loves me more than he ever thought was possible to love a person, that he was happy he met me at the stage in his life when he had "finally grown up", and that more than anything he wanted to make me feel loved and happy. From the onset, he maintained that he was totally honest to me, and totally loyal. He "would never even think of anyone else". I bought it all, although my gut kept telling me that something was not quite right. We lived together (I really did not want to get legally married again, and felt like our relationship was like marriage, as did he) but did get legally married a few years ago. However, it turns out that one year into our relationship he actively cruised for sex online and found a guy to have sex with (one time, one year into our relationship). He says it was a mistake and a lingering result of his past addiction to sexual stuff whenever he felt stressed. Upon confessing to this (under duress, since I had found evidence on our computer that he had been looking for female prostitutes just prior to my business trip out of town and just a few months after we got legally married) he was extremely remorsefull, cried (hard for him), asked for forgiveness, and swore that nothing like that would ever happen again. Nearly a year later, this time he was on a business trip, I again encountered stuff on our computer that showed that he had been looking for sex (guys this time). He swore (and I actually believe this one) that he didn't do anything, just looked online. This was followed by long conversations etc., and he finally understood that even oggling and imagining sexual encounters with others is disrespectful to me and to our relationship. That last conversation was two years ago. My trust for him is still not there. I know that he does not oggle anymore, he does not read sexual stuff online or even in the personals of free papers, and he has not "done" anything except for that one thing one year into our relationship. Yet, our relationship is not stable. I keep feeling like I need to keep the "door open" in order to leave anytime. I feel that he needs to do concrete things to make up for the pain he brought into our relationship. What should I do?