Here's the deal on taxes. If you live in DE and work elsewhere, you get credit for income tax paid to other states taken out of what you owe DE. If you would still owe DE more, I believe you have to pay the overage. But you do NOT have to pay twice.
When I lived in MD and worked in DE, my Delaware income taxes were higher than my Maryland STATE taxes. MD gave me credit for the DE taxes paid, so I didn't owe Maryland state. But the problem is that each COUNTY in MD also has income taxes that are typically equal to an additional 50% of the state cut. And the counties do NOT let you apply any excess credit paid out of state, so it is just lost to you.