Do you want this marriage to work? Imagine the scenario with no kids. Do you still want to be with him? Are you good for each other? Do you make each other happy? Even with all the niggles or annoyances does your relationship work? Are you sacrificing who you both want to be for the sake of the marriage? After you've made up your mind on these issues and come to either a painful or happy conclusion then you could consider these two scenarios:
1 - Marriage still worth it. Start fighting. Contact the other woman and warn her off. Tell her you have access to her communications. *Find out who she is ( is she married) etc. get leverage and get ready to fight dirty if she won't sod off. Start that gym routine , cut back on eating crap, throw away your naff clothes and buy new nice ones each payday ( one at a time), stop watching tv, start cooking adventurous food, organise friends to come over, plan nights out to places you've not been ( crock pot picnics, star gazing, local college sports events and so on) , smarten up the house one item a day ( paint a door frame, put up a framed pic, dejunk a cupboard) and if you can - and haven't got one - get an interesting job either by volunteering or applying with cvs for all sorts of stuff you'd never considered*. Do this and IF your marriage is salvageable ( ie he is just fooling around for adrenaline kicks ) you will have got yourself into a good place to talk about it. In short make yourself the better you ,for your own sake ,and the side effect will be a better two.
2 - Marriage failed a while ago and is not satisfying. Plan the painful withdrawal, financial advice sessions, consult legal experts, proceed from there. Work out what you want and what you can live without before the emotions get too intense. Where to live, how to budget, what child support to get, what debts to trade / offload and so on. If the marriage is over you no longer need to sacrifice yourself for him. Go back to the starred section in 1 and get started. Being a single parent is a personal war with amazing prizes for success but high costs whether you decide to win or surrender. How will you cope mentally? Are you a fighter ? Do you avoid stressful situations and does the thought of doing this make you feel queasy and powerless? If so plan a long game and don't rush to divorce ( ie get a support structure of friends, a job, a little bit of money in place). If however, you never go down for the full count of ten then you may wish to go faster.
Whichever you choose know that the kids will survive and as long as you don't self destruct ( either within a soul sucking sham of a marriage or in some singleton emotionally spent victim state) then they will thrive. Document everything you can , be official about it, send it to a private online Dropbox account or to a private email account. Scans, copies, receipts everything. If your marriage is sound and saved burn / erase in a few years otherwise save for the financial side of splitting up. Kids just need to know its not their fault and that you aren't going to leave them. Alcoholism, drug taking, over eating, excessive tv watching are all forms of escapism that can 'leave' family members behind - if you are at risk of any of this put a plan together to handle it.
If it does break up, don't limit access , don't bad mouth, don't hide all the tears and be truthful without point scoring with the kids. Accept his right to determine his own life and ultimately live with someone else. Do this and you'll go a long way to accepting your own right to choose.
Best of luck in this shitty situation.