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It's obvious that he's being dishonest with you and if you are feeling very strongly that you should end things, and you have proof that there is something at least mildly inappropriate going on, then you need to go with your intuition because although things may not be on another level now, that's not to say that it's not headed in that direction.
If he was just talking with her about their children that would be one thing but even everyday is a bit much especially when you've seen texts from him telling her that he misses her and his kids. That doesn't sound like nothing. You're absolutely right and justified in your feelings and wanting to leave. You deserve better than being lied too.
Thanks so much, I've struggled for an answer for so long, I found out recently that they are talking morning, noon and before they go to bed. It hurts but I must go on. Thanks so much!