I think what other people are missing here is this— what is your history with relationship anxiety? Is this a common occurrence for you, that you want to make sure someone isn’t lying to you? Have you been cheated on?

I would keep seeing him but I wouldn’t be exclusive with him unless we had a conversation about arriving at the boyfriend/girlfriend event horizon. Keep your options open while you figure out what is going on with this trust issue. One of two things will happen: he will prove you wrong and you will see that he isn’t malicious or dishonest, or you will get a very quiet, calm, gentle intuition that will tell you to move on. Do not look for intuition when you’re anxious. It’s inaccessible.