My boyfriend is texting his ex. What do I do?

Hello there,
I’m in weird situation. I’ve been dating for 8 months this guy, who has been super nice, supporting and loving with me. We have different nationalites and he is younger than me (idk If this information should be useful or not, well now you know).
Everything was nice and well until one day he called me super drunk, even though I asked him not to drink too much because he really can’t handle alcool, and I had to take him home while I was out with my friend. All this was quite miserable and humilating, since he didn’t do the only thing I asked him not to do and I literally had to carry him home, late at night. While doing this, I had to call his flat mate to check if someone was home, and he gave him is phone. I found out he was texting other girls, which were clearly interested into him. I got mad, and when I asked him why he was doing this, he said, still drunk, that was because “they like me”.
We passed a lot of time quarrelling about this, later that week, where he said he was sorry about causing this mess, and that he stopped texting those girls.
My trust in him though was broken.

Because of this, one day I took his phone to check if it was true. I feel ashamed writing this, because I do not think that it’s a nice thing to do, but my thoughts and my fears were stronger than my self control. I found out that his ex-girlfriend, with whom he has been for 11 years, was still contacting him. She cheated on him, they had a toxic relationship (as he said) and when we started dating, he told me that sometimes she used to write him saying that she missed him, but he told her that now was dating another girl. He said they stopped talking, but I saw the conversations. She used to text him trying to bring up the past, sending old photos and videos, and he used to laugh saying yeah we were like that, but whenever she crossed the line saying something more, he said that now he found someone he truly loved (me). If this is true, why does he keep answering to those messages? Why did he lie?
I saw that sometimes he used to send her memes or pictures about things that reminded him of her. Yet, while speding time with me, he has always been caring, loving and helped me with a lot of things.

When I tried to bring up the conversation, he lied again, saying they were not texting anymore. And also got mad, saying I don’t trust him. He doesn’t know I checked his phone.

I do really love him, but I already told him that to me this is not acceptable. Someone who keeps going back to the past is not ready for the future. Last time we brought this up we had a bad discussion, he said “Am I not doing enough for you?”

What should I do? Should I tell him what I did? What I know?

I don’t know why they keep being in touch, but I know, that if she could, she would date him again, though he clearly said he doesn’t want it.

I’m very confused and depressed about all this going on.