Oh my. <HUGGGSSSS> to you.
I have been there. From your post, there is no telling, because there are so many circumstances in play. Before you do anything, please take some time to stop and think. What do YOU want? Do you love this man, and when you think of your life in 10, 20, 40 years, is he always in the picture? By the ages of your children, I imagine you have been together for a long time, so obviously there is a lot at stake. But do you even WANT to keep him? Are YOU in love, or at least in-like. And if you do want to save this, is it because that is what you really want, or because the change is scary?
Now is the time to be very honest with yourself. It is also the time to be selfish. I mean that. Don't worry about the children, because they will be better off with divorced parents than parents who are in a toxic marriage. If you love this man and want him to remain in your life for that reason and that reason only, then it might be worth it to try to save this... but if he is coming from the position of "I'm not in love with you any more," you are fighting an uphill battle, because he clearly has unrealistic expectations of what love is.
Only a very very lucky few still feel "in love" that many years into a relationship... it's a chemical thing, it's a boredom thing, it's a growing up thing... If he's looking for tingles and rainbows and butterflies... he's not likely to find that in a long-term relationship... all long-term relationships take work, and that is what marriage counseling is.
A counselor will ask him to try to have more reasonable expectations of the relationship.... you are the mother of his children, his partner of the last XX years, and the fact that you would even consider continuing the relationship is a testament to what kind of partner you are. With all of that going on, it's a bit unfair to expect you to be the fairy princess of his dreams at the same time...
BUT... the counselor will also try to assess what it is he is "missing" and after trying to find a realistic level of expectation, ask you both to try to find a way to get there. I can tell you it is painful and unfair-feeling when you have just been betrayed by your partner to then have to hear about all the things your aren't and how *you* should be different...
And that is really hard. Because you're already feeling down on yourself. You have already lost the promise he made, to keep himself only unto you, forsaking all others. And I can tell you, even if it was all perfect and the counseling works, and you have a renewal of vows... you will never get that original vow back. No matter how this plays out, you will go through a mourning period over that.
If you love him, fight for what you want... but do it for yourself, because we all deserve to be loved for the right reasons. Don't settle for him settling, because that is a recipe for future trouble. And if you don't see him in your future (the real him, flaws and infidelity included-- not a continued infidelity, but realize you'll never forget this, forgive maybe, but you'll never really forget) then open yourself up to the possibility that maybe there is something else out there waiting for you.
If you go to counseling, it will be a lot of work. It will be compromise, which is always hard, but doubly hard when you are healing from a betrayal, worrying about the future, and mourning the loss of what was. You will learn about communication, you will be encouraged to date each other... and it may or may not work.
As for the expense, it depends on the route you take. Going through a church will be less expensive, but they will push for a reconciliation at all costs-- meaning they'd rather see you unhappily married than happily divorced... If you go through secular therapy, it can be expensive, but it's possible your insurance will help if it is billed as "therapy."
I'd really suggest you go to therapy for yourself in either case. You have been hurt, your trust has been violated, and you are probably feeling very threatened because your future no longer looks the way it did before this. Please don't rush into anything. Don't rush to reconcile and don't rush to divorce. And ask him to do the same thing.
Both of you need to do some soul-searching. It's not as simple as "Am I starry-eyed in love with this person and I want to marry them all over again?" There's so much more to a long-term marriage.
I urge you again to think of yourself. Your children will always be your children, you will not lose that, even if you lost the house or (God forbid) custody. Your job is to be the healthiest mother you can be to them, and only you can figure out what that means.
I wish you the best. This is not an easy thing. Trust your heart, but use your brain!