Cheating Partner

Me and my partner met in February of this year and hit it off straight away, everything seemed to be perfect, I've never met such an amazing, supportive, caring and respectful human being. Everything was going well until July a week before he was due to go on a month long holiday across America (which was booked prior to meeting myself).
It was a Sunday morning my partner was in the shower and I was still in bed, his phone was on the bed side table and lit up. I briefly saw a message from a woman. I decided to look at it a little further. without unlocking his phone I saw the message said "I had the hottest dream about you last night, I have to see you before you leave".
I was heartbroken but at the same time it didn't say anything to say something has happened, but he has obviously been talking to her otherwise she wouldnt know that he would be leaving a week.
I decided to confront him once he got out the shower. Straight away he promised me he had never been there. He didn't get defensive, didn't attack me for looking at his phone (which I have learnt from the past cheaters do straight away).
I asked him to block her on everything and he did, no questions asked. He even deleted his instagram account. This was the only message I saw from her I did not see what else they had been messaging. But I did learn she is a married woman with two kids.
We moved on from that until a week later he left for America on August 1st. everything seemed to be okay, he would message me all the time, we would have an occational phone call, everything seemed normal. Until August 22nd, 2 days before he was due to come home. On his last leg in Vegas. I noticed he was online so I messaged him asking what he was up to... I got "going to bed"... I asked what time it was... "late" and the conversation ended. This caused me some concern. I didnt hear from him until I picked him and his best friend up from the airport on the 25th. His best mate was the one to tell me what time their flight landed.
Since he got home everything has been going well. I completely forgot about the fact he ignored my message for 4 days, just because I was so relived his was home.
Until last weekend. 24th of November. He was in the shower, my gut told me to look at his phone (which I have never done before). There was no messages from girls since August. I found the messages from the woman who is married. Although there was nothing to suggest the had actually seen eachother, he didn't put a stop to her send nude photos. One response was " I like that".
I then looked a little further to find a message from August 22nd. He messaged a girl from America "did you get home okay?" and that was the end of the conversation. Again nothing to suggest something actually happened.
When he got out from the shower. I broke down and told him I had made a huge mistake by going through his phone. He continually said I have never cheated on you. I asked him about the sexting that was happening while we where together, he sort of worked his way around that question and I didnt get an answer. I then asked about America, he then said I could ask any girl that he was on the contiki tour with and they would tell me nothing happened. But the problem is, He wasnt on the contiki at the time of the message... and she was from America.
He then got upset saying that he hates that I can't trust him. and that Im pusing him away. after a few hours we where back to normal. We spent the whole weekend together.

Its still playing on my mind. He still leaves his phone alone in the room with me (which kinda shows me he doesnt have anything to hide). From what I can tell if he did cheat, it hasnt been since he came from America. Could the America trip have been a last stint before he settles down, our should I be concerned. I dont know if I should end it, Im scared, I would like my best to work things out.