Did he know you were looking through his phone? That's my first concern. Snooping on your partner is not ideal. At the same time, I don't think partners should have secrets that they need to deliberately hide from each other.
If he knew you were looking through his phone, and he deleted the texts to keep you from seeing them, then it concerns me that he's being dishonest.
If it's true that he's basically just embarrassed about what his texting conversations, then perhaps it would be worthwhile to talk to him about it. Let him know that honesty is more important to you than propriety.
At the same time, if he wants those messages to be private, than he and Ms S have the right to their privacy. At that point, it's not lying and secrecy. You may need to accept that there are some aspects of their relationship that are separate from you.
If I wanted to be labelled, I would hang out on grocery shelves.
Gralson = my husband, Auto = my girlfriend