**UPDATE** husband has emotionally cheated on me 😭 (sorry for the long post)

We've just had our first baby 6 weeks ago and have been married for 4 years. We were together for 5 years before that and I never ever thought he could do such thing as we went through hell to be able to get married (families opposed etc).
However, I noticed the name of a girl that kept popping up on his Snapchat. I had a gut feeling that something wasn't right but I gave him the benefit of the doubt. Then one day I noticed she had a "yellow heart" next to her name. For whoever doesn't use Snapchat, this heart comes up when you frequently Snapchat someone a lot, so I asked him who this person was, he said it was one of his waiter friends (he owns a restaurant). I also noticed his behaviour being strange as he kept hold of his phone a lot more and never used to hand it over to me as easily as he used to. 
Fast forward to this morning, he was asleep and I decided to check his phone (I know it's a sly thing to do but I had to do this for my peace of mind).
I went to his chats and this girls name was at the top again, I opened the conversation to find a picture of her lingerie !!
I couldn't believe my eyes at first, then as I scrolled up there were several saved messages in which I soon realised that he had been sexting her.
She even asked him at one point if he's single, to which he told her that he's married but that I was asleep as it was late night.
I also found out that he was the one to initiate this whole thing! As he admitted to flirting with her when she came to his restaurant as a customer.
I confronted him about this straight away and he admitted to it although he doesn't believe he cheated as he says he was never physical with her.
I'm sorry but first of all, how and why should I believe him when he says he was never physical?! And to me sexting someone is also cheating!!!
He's been saying sorry and even started crying about this. He keeps saying to give him a chance and that he'll never let me down again and he's learnt his lesson.
I asked him why he did it, and whether he's unhappy in our marriage but he said there is no problem and he's really happy. How can it be? Why would he go elsewhere??
I know I have become less attractive since being pregnant and having a baby. I have stretch marks everywhere and I teared badly during labour which still hasn't healed, meaning we haven't had sex in almost 2 months. I know that it must be hard for him but I never thought this would happen!!
Also, they first ever started messaging each other last December, meaning it was around the time I conceived! It also means this has been going on for over 10 months!!!
He keeps asking for another chance but I don't know what to do. We have a 6 week old baby and I don't want to raise him without a father but at the same time I don't wanna be the fool that sticks around and gets taken advantage of again. Any advice would be appreciated. Am I overreacting? 
**UPDATE**
Wow. Thank you so so much to everyone for taking the time out to read my problem and give me advice. I didn't know who to go to, and posting on here was the best thing I could have done =)
So I have spent some time to think about everything and have decided to give him another chance. Just ONE and I've made sure he's aware of that. I've told him that if I ever find out anything again, or if I see him flirting with anyone at all or doing anything wrong again, I'll pick up the baby and disappear to the point he won't be able to ever find us again.
He keeps apologising still and says I won't regret it. He says he will make sure it never happens again as he never wants to hurt me like this again.
I've asked him many times as to why he did it, but he says that he honestly has no other reason than the fact that the girl was "easy" and he just enjoyed the attention. He is adamant that there is nothing lacking in our relationship and he's 100% happy. The problem is with his character and the fact he couldn't stop it all once it started, as he got so involved and enjoyed all the attention too much.
I've gained full access to his phone now. My fingerprint is recorded, I've deleted his snapchat account completely and I've gone through everything in his phone for my peace of mind. Even in future, I told him I'll keep checking his phone and he needs to be okay with that. I also have his Apple ID password so I can track his location through "find my iPhone" whenever I want.
I know it won't be easy, but he says that he's willing to work as hard as it's needed to gain my trust again and for us to be a happy family.
I'm gonna make him work for this. He's living at home with me now but he's not allowed to come near me or touch me. I'm not talking to him properly although he keeps trying, and I won't be okay around him again until I feel comfortable and he needs to be okay with that.
He says he is determined to do what it takes not to lose me. We'll just have to see now I guess. He'll really have to prove himself.