To start a foreclosure, just miss a couple payments. They'll get the message. Might try restructuing...
In a short sale, you may not be eligible for the Fed Income Tax forgiveness, and the bank could come after you for the "shortage".
If you let it foreclose, the sale of the home satisfies the first mortgage (at least), and the bank sends you a 1099A/C, which is technically a capital gain. Recent Fed actions may give you the chance to be forgiven on this "ghost" income.
A second mortgage (80/20 maybe), could convert to unsecured debt, and the bank may want that. Or maybe 1099 you again (hope for this one).
Google these: Non-recourse, deficiency judgement, foreclosure forgiveness, primary purchase funds, abandonment, willfull destruction.
Sounds morbid, but: Make two budgets, one with struggling to make mortgage payments and the other with NOT making the mortgage payments. Go 4-6 months out, and see what's left over. Around the 4-6 month, that's when you need to pay rent/deposits on your new rental home. (you may be done hoping for the best, but now you need to plan for the worst -- Planning beats hoping every time.)
Walking away brings shame, but you planned and took action. It's your family against the world; you lost these battles, but you intend to win the war.
Your new rental home: two-year lease, $500 cheaper than your mortgage, and rental insurance. You are living on cash flow now, so an emergency fund is mandatory.
Next: No payday loans, no unsecured 35% personal loans, etc.
Next: Credit union, secured credit card for $1000.
Next: No new "garbage" credit for 18 months, just wait to see what happens with the Credit union relationship.
Just went through this, it hurt, but it was necessary.
Good luck!
