Now, I’ll disabuse myself of asking how it is you’ve discovered his online behavior or the content of his texts and simply say that it’s obvious that there is a trust issue there and it will not just go away. He will not stop his behavior until he feels fulfilled either within or outside the relationship. Your comfort and security in this relationship are likely to be compromised as long as he persists in doing something you’ve asked him not to do.
As for how worried you should be, I’d say it’s reasonable to have concerns when your spouse persists in a secondary relationship that he keeps secret from you. From the context of your letter it sounds like you “discovered” this communication rather than him divulging it and that he-in anticipation of this discovery had been editing his communication. But this is not cause for worry- it’s a call to action. You either tolerate this or you don’t. If you are comfortable with it, and you trust him you stop worrying immediately and never peek at his phone. If you are not comfortable but are committed to the preservation of your union you remind him of the vows you took “forsaking all others” and ask him what it is he needs that he’s not getting that's causing him to forsake you – and then decide if what he needs is something you want or are able to provide. If you really want to work this out It’s important that you create a safe environment for this discussion – he’s hiding because he fears judgment and retribution. He’s looking for a trap. Remind him that you gave your all to him and that just like he wants you to come to him first for whatever you need, you want the same from him. I wish you all the best.