As CarolA correctly points out, it's no big deal for an airline to file Chapter 11. Remember that United has been operating in chapter 11 for over a year now and most other old-style airlines are in much more difficult financial straits thatn Delta. The main problem Delta has is a Very Expensive pilot contract, while all the other Delta employees have made big concessions.
If they change a few flights, ticket holders will still be OK, they just might have to change their itineraries a little, and make sure that their seats are the ones they wanted. I have two trips booked on Song and fully expect to make those trips as expected, something I would not be as confident about if I were flying AA, USAir or United.
Bottom line: don't worry! Even if they do declare chapter 11, they are gonna keep flying where we need to go.