Deceived,
First, sorry you have joined the ranks, but you will get plenty of support here. I know your head is spinning, but I hope the following advice is something you can take to heart and use it for your healing.
First, you need to find if the sleazy other man (OM) has a wife. If so, you really do need to tell her. Your wife has already proven that she is a first-rate, look in your eye, liar. Called it off?? I doubt if she has done anything but take it deeper underground. I realize you want to believe her, but just look at her recent actions. Let the OM's wife know about the affair in a detached, but respectful manner, BUT DO NOT TELL YOUR WIFE THAT YOU ARE EXPOSING. If she is still in the affair, she will resist with everything she has to let the other wife know about her adultery.
Next, you really do need to see a lawyer. I know you will think this is counter-intuitive, but your hand is weak. You likely do not know what a divorce will entail. Well, I can tell you - it will involve you being screwed over by your WW if you are not careful. Talk to the lawyer, learn the process and learn what it takes to protect you and your kids. I would even suggest having the separation agreement drafted and in your back pocket (with your terms). Your wife likely thinks she has you under her boot and that you will do whatever it takes to get back with her. Too many WWs do. Well, your best defense is the knowledge of what a divorce will entail and your eventual acceptance that you can live without sharing her with another man. Knowledge is power and you need it sir.
Next, get STD testing. Sickening, I know, but you must protect your health.
Next, I highly advise that you stop marriage counseling and seek IC for yourself and demand your wife seeks REAL IC to fix her brokenness. Marriage counseling with a cheater will likely only frustrate you. All the problems will be laid at your feet because you are the man and her affair will likely be rugswept. She had the affair and SHE has bigger problems than the problems in your marriage.
Finally, detach. You most likely will desperately want to fix her. That is what guys do. You will think you can lead her out of her soullessness. You cannot fix her. You MUST detach from her and her brokenness and hatred. You need the detachment to have perspective, because perspective is needed to make the best decisions on your behalf and behalf of the children.
Strength to you sir. You will need it!