I am very sorry to hear about your situation. At times like this the stress you are under may make it difficult to make the best decision for your family. Knowledge is power, and right now you need someone that can provide you with options and their pros and cons, then you make the choice.
First, what would you prefer to do - stay in the home, or move?
Second, if you wanted to stay, do you have the income to pay a mortgage payment?
Third, if you wanted to move, do you have the cash to pay first and last month's rent?
So let's take these one by one:
If you want to stay in the home, you need to try to negotiate with your lender for a loan modification. Most lenders are open to modifying your loan, but only to a point. One proposal I recently read about was a loan that would have a payment no more than 38% of your income, and 90% of the value of your home.
You could find out what the current market value of your home is, and have a Realtor compute the numbers for you.
If you want to move, then you'd need to show "hardship", which divorce certainly qualifies. In the "hardship letter" you explain how you were able to make payments at one time, and now, due to the divorce, are unable to make the payments and do not see that picture changing in the near future. They are going to ask for a financial statement of your income, assets, and liabilities.
Note that a loan modification and short sale process are two opposite paths and probably do not mesh well. I recommend talking with a Realtor who can analyze your situation and help you work out alternatives.
So far as the decline in value of your home, I would not focus on that issue. Many people have had the market value of their home affected, but the majority will just have to keep paying. If your home went UP in value, would you give the lender the extra money? Of course not, so when the value drops, same rules apply.
I would be happy to refer you to an expert in short sales in your area. Just contact me via my Trulia profile.