I always say, listen to your gut instinct.

If you smell a rat, there sometimes is ...

Here's the catch. You obtained his texts underhandedly (rightfully so!) ... so how do you bring up the fact that you know about them?

He is out of line and should under no circumstances be discussing your relationship issues with his ex - it is absolutely none of her business.

You could try ask him, "Who do you speak to when we have problems xyz. I've spoken to my best friend _____ about seeking a job / my previous job issue as well as you."

See if he brings her name up. If he doesn't you have the option to tell him you know - but be warned that is not going to go down well and likely would have consequences.