Hi! Short answer, I do not know. Anyone who suggested they do. Is it possible to find out? Ya sure. You asked what to do. If you care enough, if it is worth your time - Find out. If it is not, well leave (When you can).
Long Answer:
Here is what we know:
I've always had problems with him flirting and texting other girls.
I do log in to his social media because i do not trust him at all.
He's direct messaging with some girl I believe works at the gym.
Of course, he's already deleted everything they first talked about,
***dirty sex talk redacted "________" "____kidnap_____"
she has a boyfriend and he’s good to her.
he’ll just find a way to hide it.
We do not really know a lot. We know your husband is a liar. We know you have proof or genuinely believe it. You did not wake up one morning and say, "Oh boy that would be great if i thought my husband was a liar". If you lack proof that instinct you have is typically enough. Likely have been picking up pieces which lead you to believe this. Or you are just nuts & think everything is a sign..... We know he has no issue with talking to other woman sexually. If I am talking about kidnapping some chick, or if she is dreaming about me kidnapping her - this ends kinky. Need to be honest, right? We can linguistically dance around it, "He is being cute, playful, harmless flirty boy" sure. But it is sexual. Is that ok to you? I assume this is not ok with you? With my partner and I, we usually wind up sleeping with this 3rd person together, so it is fine for me (or her) to engage in such talks.
Why don't we find out if he is cheating. Figure out who he is talking to. "Someone at the Gym" is not acceptable. If i asked you to get me specific historical data. I would expect you to be on your A game.
Pipl.com
Spokeo.com
Socialmention.com
Social-searcher.com
Peekyou.com
He knows you know his social media accounts. Of course he has others or is he really stupid? So we are looking for information on both. Boolean searches are as important. Plug in all information you know about them. Come up empty use other ways to look.
SMS messages, photos and cache from social media is really never deleted from mobile devices. And never from computers unless the person is acting with serious intention:
http://www.decipherforensics.com/android-forensics/
While you are not allowed to. You could use google to track his phone, if you have the log in. I am saying this so you do not do it, you are not allowed to. I do not like these write-ups. But this is a good intro:
http://trendblog.net/how-to-track-your-lost-android-phone-without-tracking-app/
Rip out someone's heart for me and I will tell you their bloodtype. Get access to their mobile device, I will tell you anything you want to know. That is an important mantra.
I do not know how much you care, there are hundreds of things to do. If you do not care leave him or stay and cheat too, it will genuinely make you feel better and might help (really). If he or the relationship is worth it. It starts with the last 6 months of loud discussions, their cause, the tone and outcome. Since we are curious about him.... And this is cheating his use of Distraction, Deflection, Denial & Projection. You never told us about them though... His habits changing? How's intimacy/distance? Why do you feel that way? This is not hard, but it takes time and effort. It has to be worth it. There is just too much you did not tell us (as there is not enough room).... Possibly all this typing helps someone one day.
**Are they chatting or cheating, if you believe one or the other you are probably correct. :/
GL ask more later, I guess