Am I overreacting? Hiding texts with who I assume to be other women

My boyfriend and I have been together for almost three years, and up until recently, I would have said we had a pretty solid relationship. We live together, we have talked about getting engaged eventually, and I have never considered myself the type of person who checks phones or constantly worries about cheating.
A couple of weeks ago, I noticed he changed the password on his phone. I only realized because he asked me to check something while he was driving, and the password we had both known for years no longer worked.
When I asked about it, he laughed and said someone at work had figured out his old password and kept trying to mess with his phone. It sounded believable enough, so I let it go. I did notice that he did not tell me the new password, but I also did not want to act entitled to his privacy.
Since then, though, he has been acting differently with his phone. He takes it everywhere, even if he is just going to the kitchen or shower. He keeps it face down whenever we are together, and he has started turning the screen away from me when he is texting.
I tried not to overthink it because I know phones can become a sensitive subject in relationships, and I did not want to create a problem where there was not one.
Then last night, we were sitting on the couch watching a movie when his phone lit up on the coffee table. I was not trying to read it, but the screen was directly in front of me, and I saw a message that said:
“Can’t wait until Friday” and then something else.
I did not see the name because he grabbed the phone almost immediately and turned it over.
I asked him who it was, and he said it was his cousin joking around. The explanation felt strange because I know most of his family, and I have never seen any of them text him like that. I asked why his cousin would send a heart and say they could not wait until Friday.
He got defensive and said I was reading too much into a random message.
I told him I was not accusing him of anything, but the situation looked suspicious, especially after the password change and the way he had been guarding his phone lately. I asked whether he could show me the conversation just to clear things up.
He refused.
He said that once I started asking to inspect his private conversations, it meant I did not trust him, and that showing me the messages would only reward controlling behaviour.
I understand that people deserve privacy, even in a relationship. I do not believe couples need to have unlimited access to each other’s phones. But I also feel like if my partner saw a message like that on my phone and was genuinely worried, I would immediately explain it because I would not want them sitting there imagining the worst.
After that, he barely spoke to me for the rest of the night. This morning, he said I owed him an apology for accusing him of cheating. I reminded him that I never directly accused him. I said I was uncomfortable and asked for clarification.
He said that was basically the same thing and that my insecurity was becoming exhausting.
Now I am questioning myself. Maybe the message really was harmless, and maybe asking to see it crossed a line. At the same time, I cannot ignore how secretive he has suddenly become or how quickly he shut the conversation down.
I have not asked to check his phone again, but I also do not feel reassured. I feel like his reaction made everything seem even more suspicious.
Am I overreacting, or would most people find this behaviour concerning?