Um, why are you considering marrying this person?
He has cheated on you. He lied about how much he cheated on you. He actively reaches out to other women on the internet and doesn't care if you see it. That isn't even respecting you, let alone showing any real commitment to you. I would be shocked if he DIDN'T cheat on you even more than what you know about. I know you say he "used to" add and send his number to random women, but if he's still looking at profiles you can bet he's still sending messages and numbers out.
Guilty consciouses pick a lot of fights over nothing. It's a way to deflect from what they're doing.
I know it's scary to leave a long term relationship, but there is nothing healthy about this situation. A ring on his finger won't change those behaviours.

He *will* still pull this crap after you're married and it will be even harder to remove yourself from it at that point. I know it's even scarier to leave with a baby, but again -- it isn't something that's going to change with marriage.
By staying, you not only send the message that how he treats you is okay and you will put up with it, but you send the message to your children that that is a normal and okay adult relationship. You send the message to any daughters that it's okay for a man to treat his partner this way. You send the message to any sons that it's okay for them to treat women that way. On top of the fact that you don't deserve that treatment and you owe it to YOURSELF to remove yourself from it, you also owe it to them to give them a stable and healthy environment to grow up in.
If he showed any signs of being willing to change, I would still advise you to hold off on the wedding but perhaps get into some kind of therapy. The reality sounds like he doesn't think he's doing anything wrong and he unabashedly continues the same behaviours right in front of you. That's a huge, huge, huge red flag.