I am so sorry you are dealing with this, Nicole. Soooooo very sorry. Your h has demonstrated some very immature and questionable behaviors before this. In my book, what he did is cheating. No question about it. It's a total betrayal, and don't let him gaslight you into believing it isn't. This is serious, and it isn't going to change on it's own.
If you can muster up the strength and bitchiness, kick his ass out temporarily. I know how hard that may be, and I know it may be impossible. But he has family there; have him stay with them. In my mind, there is nothing wrong with telling his mother exactly why her son needs to come home for a while. If you can't do it, can you at least get him on the couch??
Next, get an appointment ASAP with a marriage counselor. Do you have health insurance? Do you have EAP benefits? Most jobs do. The EAP benefits will provide 7-8 free sessions. Completely free. Insist that he goes with you. That should not be an option if he wants this relationship to continue. If he refuses to go, go alone and get the wisdom the therapist can give you about the situation.
You guys are young. I was there once. I got married at 21, my xh was 22, and then we had dd1 within 6 months. It's hard. If there are not some serious changes made, he will never change. Oh, and he has lost his right to privacy at this point.
Let me tell you what happened to my ex... you can use it as a warning for your h. He lost his family. He lost the wife he loved, he lost seeing his kids everyday, he lost having a real home and family life. He lost money. He had to come up with a ton of money for a lawyer. And he pays a ridiculous amount of child support to me. Plus healthcare for the kids. Plus half of out of pocket expenses. And guess what? You can't deduct child support. So the payer is taxed on their entire income, even though they are giving 1/3+ of their pre-tax income away. Yep. Sucks, doesn't it?
Also, did you take screen captures of the texts? If not, do it before they are gone. Are you sure it didn't go further than texts? I hate to say this, but meeting at the bar sounds like he had more plans than that.
I'm sorry if this all sounds crazy. I have some experience, though. My first goal for anyone is to save the marriage. But you also have to make plans and be safe in case that isn't possible. Super hugs to you. Keep us posted, ask anything you want, feel free to pm me.