How should I file if my employer did not change my address after I moved from DE to MD?

In general, what your employer puts on your W-2 does not determine how you should file.  If you lived in a state for only part of the year, then you would file a part year resident return for both states, even if the employer did not change your residency as you will still have a tax liability to both states. 

The most important thing on your W-2 when it comes to your information is your SSN and your name. 

Which state did you work in?

What you will need to do is start in the personal info section and tell TurboTax that you lived in another state.  Then after you finish the federal, you will file DE as a part year return, walking through the steps, you will be asked how much of the money was earned while a resident of DE?  If you worked in DE, then ALL of your income will be taxable to DE.

Then you will do the MD return, you will again be asked how much of your income you earned while a resident of MD.  Then you will only be taxed on that amount (unless you worked in MD the entire year, then you would be taxed on the entire amount) 

Basically, whichever state you worked in for the entire year, is the state you will want to file first as that is the state that all of your income is taxable to.  Then when you file the other state that you did not work in for the entire year, you will apportion your income while a resident of that state and then you will take a credit for the taxes paid to the state you were working in but not living in. 

Filing Multiple States