Bankruptcy Law Questions? Ask a Bankruptcy Lawyer.
The banks will most likely get what is called a "deficiency judgment" against you. That is the difference in what you owed on the note, and what they got for the house after they sold it at auction. Those judgments are civil judgments and can be executed on just like any other judgment. They can lien your property (maybe not your principle residence depending on where you live), garnish wages, bank accounts, intercept tax refunds. The Judgments can remain indefinitely on your credit report if they refile the judgments every 10 years. They will remain at least 10 years on your credit report.
If you declare bankruptcy you most likely will be able to keep your home, but you will have to include all creditors including any bills you say you want to keep paying. You cannot take partial bankruptcy. Dependng on your situation and income you would have to consult with an attorney to see which Chapter (7 or 13) would provide you the best options.