Hi Layesh,
I'm really sorry to hear that this has happened, and that you're having to struggle so much. I've had my own marriage fail because he "fell in love with someone else," and things became complicated for about six months after that- during part of that time, I definitely had a lot of these issues, such as imagining them together, and feeling very hurt and angry, and confused.
I have to ask... how did you find out that he slept with this person? If he confessed, do you know why it took him so long to do so? How does he feel about your relationship, and about why he felt he could sleep with this other person?
In other words... have you sat down and had a serious conversation with him about it? Having more information might help you to become less confused. Especially knowing what's going on in his mind about your relationship, and why something like that happened, may take the decision-making for your future out of your hands. As hard as it is to deal with for so many of us, it takes two to tango, and if he's not on-board, you're both going to have to deal with the consequences.
However, if you find that he has no issues with how things are right now, there are clearly issues on YOUR end, and they MUST be dealt with. Please, NEVER think that your needs, issues, and expectations are any less important than anyone else's in your life, including your children or your partner. If you aren't getting what you require, and you simply roll over and take it... well, that sends a message to you, and all the people around you, and it is a lie. You are SUPER IMPORTANT and NO ONE should EVER say any different. EVER. If they do, you're going to have to seriously re-evaluate why you're around that person, and whether what they contribute is really worth you being put down and insulted that way.
I realize that forging ahead and being on your own, with two children, is terrifying, but I think that the main thing you must keep in mind is YOU. There are so many ways for a single mother to get along in the world, and part of that may be something like joint custody- that way, you are not needing to raise your children alone, or pay for them alone. There are options, and you must remember that.
Anyways, for the more immediate problem, there are a couple of routes to go: Staying with him and trying to re-establish trust, or deciding that this is a dealbreaker and cutting things off. Obviously both come with their positives and negatives.
If you wish to try to stay with him, think long and hard about what you need to trust him. Many people go on the route of "Big Brother" and want constant check-ins, and want their partner not to talk to anyone of the opposite gender. This is generally because of the lack of trust, and it tends to put huge strain on the relationship, and the children. This isn't what we want! You want to be able to have him go out in the world, and trust that he will NEVER do this again. And, it's you that must figure out if that trust is possible to regain, and how it might be done.
Again, I must reiterate: This is something serious. You are completely validated in being hurt, in needing to be taken seriously, and in having your needs met. It doesn't matter what kinds of excuses are given, if he was drunk or high, or if he "just wasn't thinking," or whatever. He made direct choices that lead him to what he did, and if he didn't think of you and his children, well... that also sends a particular message. That's a big reason why it hurts, among many others.
I found journaling to be extremely helpful in times of this nature, and continue to find it useful. It helps me figure out how I really feel about certain issues, and figure out the things I need, and it helps me to feel grounded and established in myself.
Good luck, and don't be afraid to reach out. The people here are amazing. Send me a message if you want to talk more privately, or anything else.