Try To Work It Out Or Just Leave?

I am trying to decide if this relationship is work my time. We have been living together for 18 months. Last year, he was texting his ex-wife (no kids together) quite a bit. I didn’t have any issue if they were and would ask what she was doing. He told me repeatedly that he never texted her. I saw phone records later that disproved that. After other issues, I moved out. Moved back in and have had further issues. I asked him to lock his phone records but don’t like that I don’t know if he is being honest with me.
2 months ago, he didn’t come home all night. Said he fell asleep at a friends house but a receipt I found showed he actually drove the opposite direction right before he would have gone there. He refused to let me see his phone texts the next day til after he deleted whatever was there.
I want an honest and open relationship. I want to be able to trust him, but don’t. I feel dumb asking for the phone records again, but don’t feel that I can trust him. A virus posed as a keylogger was on the computer. He blamed me…then went and changed all his passwords again from a different computer. I feel kind of like he is keeping me out as far as possible. But maybe I am just being insecure. Should I demand openess, leave, or just let it go?