It does sound like you are fucked. Not going to give you false hope.
What I would do is end all relationships with current girlfriend, not fair to drag her through this, relationships are expensive and you are currently broke.
Quit the my second job and use that to lower my child support.. Move into a more affordable apartment, get rid of the car settle for something more affordable. Lease and loan should be under somebody else name not your own.
Would not pay for the house, judge will have to lock me up before I do so, her and the kids will be just fine without you paying for the house. Yeah your credit will be fucked but that's the situation you are being put into, and it's an unreasonable situation so become unreasonable yourself.
Make sure you feed yourself and pay your rent first, along with all the bills that by now should be under someone's else's name as I mentioned above, everything else comes second to that. Get rid of the lawyers, stop paying your college debts.
Again very important, put everything you buy after under a parents or good friend name.
Push comes to shove and they try to garnish your check then just have your boss pay you off books, if they can't, quit that job and get a job off books. When you get paid just mail her what you can afford to mail her via money order. Keep your receipts. Don't be a dick and send her a hundred bux.
Become that deadbeat you hear so many people talk about. Fuck it what else can you do to take back control of your life and live comfortable. Relationship with kids might be tough but when they grow up they either understand or they don't, either way you need to do what you are forced to do in order to survive.
Or you can go the righteous route go broke and live like a dog in the streets for the next 20 years.