Tell her that you will not leave your home and you will not give up on her, the kids or the home.
Tell her that you'd prefer that she stay and work with you to rebuild your life, which might mean selling your house or whatever BUT that regardless of what SHE does you are moving forward.
Tell her that if she plans on violating your wedding vows then she needs to consider finding a place to live because you will not tolerate an adulteress living with you.
And yes, you are correct that any 'man' who would chase a married woman whose marriage is in trouble is garbage. He is not the problem though, SHE is in a panic about her life and running away. Forget about him because he is a symptom.
This is the REAL problem: She has not seen you be a man in a long time. And a man is 'in charge'. She doesn't feel 'unsafe' because of the financials, she feels 'unsafe' because she doesn't believe you have the solution, probably because YOU have not been expressing confidence that you DO have a solution.
The 'solution' may be 'sell the house'. And that's OK. It's the confidence and the authority that is missing.
Go read 'The Way of the Superior Man' by Deida.
Go to the 'makingherhappy.com' website and join the free e-mail list and look at the free mail archives and get an idea what 'confident men' look like. You were that man, she married that man. Now you are living in fear and you need to turn that around.