We’re trying to sell our house via short sale. If the bank will get less money than what we owe them, does the bank have any recourse against us to try and collect the shortage?

In Tennessee , the lender can obtain a “deficiency judgment” against a homeowner for any amount they are unable to recover through a foreclosure or short sale.  This means that if you have a mortgage for $200,000 and a short sale brings the bank $125,000, they could possibly obtain a deficiency judgment for $75,000.  In most cases this does require a separate action to be filed in court, causing the mortgage company to incur further expense.  The lender is also acutely aware of the borrower’s inability to pay so they often see further collection efforts as fruitless. 
 
Of course if the home goes to foreclosure, the bank retains the same deficiency judgment rights they have with a short sale.  The main difference is that the amount they seek following a foreclosure will almost always be much more than after a short sale because of the added expense and likely lower sales price of a foreclosure sale.
 
There are plenty of negatives (emotional and financial stress, credit damage, etc.) that come along with a short sale, but just about every single one of them turns much more damaging if you allow your home to be sold at foreclosure.