Sorry you are going through this.
When my husband betrayed my trust I talked to him about it and how it made me feel. I didn't give him time to try to cover his tracks, but I also didn't snoop. I just outright asked him to give me access to his accounts while we sat across from each other at the table and then went through them with him.
It wasn't necessarily the act of going through his accounts but his reaction to the request that I was interested in. If he had said no, hesitated or got angry at me for even asking or insinuating he might have something to hide, we would have had a totally different conversation- because that would mean he did have something to hide. He showed me everything and we moved past it.
No one knows your relationship or your DH like you do. I'm sure, in your heart, you know what he would or wouldn't be capable of. My DH would not have the patience or consistency to continually delete conversation history and hide his tracks.
This may not be the popular opinion, and I don't mean to offend anyone, but I personally don't think that suspicion of his dishonesty gives carte-blanche to be dishonest yourself. I would have felt horrible if I went behind DH's back and found nothing; would I tell him that I'd done that? Would he forgive me?
One last thing, when I was pregnant with my first I went through several periods where I thought DH was going to leave me because I was acting all crazy (hormones) and, well, just because he could. After talking about my feelings with him, I realized that I didn't like the fact that if he wanted he could just walk away, but I couldn't- I was literally stuck to the baby. Those feelings went away and DH was more patient with me because he knew I needed him to be.
I hope you manage to work through this in a way that you are happy with, no matter the outcome.