Well, if you peeking at his stuff was an agreed to consequence thing -- and he gave you his password to enable you to do it? And then he goes and lies? Behaves weird so you have the urge to check? And in doing so you find confirmation he was lying? More fool he.
Why bother waiting for him to come clean?
Lies are a dealbreaker for you. Could dump him NOW and move on and be free of him/this behavior sooner rather than later.
I told him about a month ago that I didn't trust him. I should have just gone with my gut feeling and ended it then.
Next time DO listen to your gut earlier -- there's no need to be having this sort of "peeking consequence" if you are willing to build trust with a person.
If something seems off to where you are not sure you want to build trust with them? That's not a "joyous yes" that you want to build trust with them. So could call it a "working no" so you can decide to move on earlier. Saves time/effort/energy. If not willing, not willing. End of story.