Do I need to file a non-resident State return if I had earned wages in that state for part of the tax year? (The W-2 does show some withholding for that state.) I have already reported the other state's income and withholding as part of my total wages from all sources in my home/resident State return. Or should I have left the non-resident State income out of my home State's return?
You correctly included all of your earnings in your resident state; however, you need to file a non-resident tax return for the other state you worked in. If the non-resident state taxes you, your resident state return will then calculate a credit for tax paid to another state. If the amount you earned in the other state is below the threshold for owing tax, the only way to get your tax withheld back is to file the non-resident return.
You correctly included all of your earnings in your resident state; however, you need to file a non-resident tax return for the other state you worked in. If the non-resident state taxes you, your resident state return will then calculate a credit for tax paid to another state. If the amount you earned in the other state is below the threshold for owing tax, the only way to get your tax withheld back is to file the non-resident return.