A lot of shaming going on here. OP should ignore all the shaming, and also if there was any positive compliments, but I don't think there was any. Ignore everyone else's personal morals. Since they are based on their own personal beliefs, experiences and desires. Life live within your own morals, not society or this message board. Seriously, none of us know you or your exact situation. What he said or she said. You never said your wife is a good woman, but other posters are somehow claiming that she is. You should only be concerned with them highlighting options or problems you haven't thought about. Other than using it as a barometer as to others could react, you should ignore their personal opinions or assumptions about you, your situation and how one should behave as an acceptable citizen in society.
If she wants you back, but you want to have fun on the side, bring up the option of an open relationship. Assuming you are willing to allow her to explore if she desires as well. It's not the socially acceptable thing to do, but it could fulfill your wants, and it may help her as well. It could keep the family together and be better for the kids as well.
You put yourself in a very bad situation. You know you screwed up. Even if she was a bad person and you wanted out, there were other ways to do it at a lower cost. Time to fix it, but with your happiness in mind. Weigh the price of each future action and determine if you are willing to risk that price for the potential happiness from each decision you are about to make. Ideally write them out, list potential results and chances of those results, good and bad. Include financial and emotional costs. Also look at how other people may or may not react and if you are willing to accept it. Basically do a risk analyze and cost-benefit.
Your life is your own and no one else. Live it the way you want. Accept all the possibilities, but also be accountable for your actions. Do not blame others for how they will react to your decisions, you should have weighed that before you made those actions and have planned accordingly, or be willing to accept the fallout, pay the price and move on.