Cheating or no?

April
I saw conversations through text and Instagram DMs between my boyfriend and another woman that had gotten flirtatious. With phrases like "I'm getting hot and bothered" and "you make me wet". Would you break up with your boyfriend if you found this on his phone? How would you react. 
This happened to me last night and the only thing I could do was leave. I came back 2 hours later and we did talk about it. He cried saying he fucked up and begging me to stay but all I could do was cry. I'm not sure if anything will be the same with him and if I will ever trust him as much as I used to. 
To make things worse we recently had our 1 year anniversary and we've lived together since the end of November. 
PLEASE HELP. 😞