Boyfriend Thinks I Cheated on Him(Gay Relationship)

This is my first time posting on a site like this so forgive me if this is a bit new. My boyfriend and I have been having some issues recently. Mainly in the past month. To give some background info, we have been dating for about 3 months and things have been moving very quickly. Things started out amazing with both of us telling each others parents/friends, going on dates and spending a lot of time together. I really believe both of us truly love each other. He is 33 and I am 20, both have previously been in relationships that have lasted several years and I feel we are both completely committed to each other. We decided we wanted to take things to the level of living with each other.

Let me start by saying I have in no way cheated or lied to him. One issue started early this month while we were both on holiday. We were both drinking quite a bit together and just enjoying our break from work/school. One of the nights ended early with him falling asleep on the couch. I was not really tired yet and decided I would go pick up another bottle of wine. I went on a walk(we live in the middle of a busy city) to get some and decided I might stop by the bar instead and get a drink. Im a friendly guy and struck up a conversation with another guy walking to the bar. I immedietly told the guy I have a boyfriend and was just looking to have a drink and hang out for a bit. We were at the bar for an hour or so and then I went back to our apartment. When I arrived home, my boyfriend was awake and concerned about where I had been. I told him the whole story and he was upset saying I broke his trust. The day after there was quite a bit of arguing and talk of me leaving the apartment, we resolved the issue and left it at me have to regain his trust. I understood the situation may have looked a bit sketchy so I told him I wouldnt do it again.(Note: Although slightly different, he has been to the same bar with friends but told me ahead of time). Several weeks later, we were in a similar situation and I went out to get a drink but by myself at the same bar. I met a girlfriend at the bar and we talked for an hour or so. When I came back, he woke up and asked me where I was. I explained and he was just as furious as before. He threatened to kick me out that night but let me sleep there for the night. The next morning we had a very similar conversation ending with him telling me I could stay.

I understand I messed up. It was not worth it to have gone to the bar either time. I was intoxicated at the time and didnt see the issue and thinking it over I may not have been comfortable with the situation reversed either but I knew I wasn't cheating and/or lying. I immedietly told him everything both times. Now when we argue, he will bring these up often and tell me he still doesnt trust me, that all he has to go on is that "I say I didnt cheat on him", and he has even said that I probably did cheat on him. There is nothing showing that I did cheat on him though. I have let him look through my computer, phone, and even told him I would not go to gay bars without him anymore or drink liqour.

Should I back off and give this time to play out and regain his trust or is he over reacting and being insecure? I would really like for things to work out but I know Im not good at being controlled/sacraficing for no reason either. Thank you