The general rule is that you owe tax to the state where you live and to any state(s) where you worked.
In your case, since you lived in DC all year, you should report your total annual income to DC.
The purpose of the credit for the other state is to prevent double taxation on the same income. But you don't have that, since, as you noted, Texas has no state income tax.
So, no double-taxation, no credit.
This is the way the system works.
In the case of the first poster, they were actually a resident (part-year) of two states, so they rightfully split the income between the two states they lived in (or used one state's tax paid as a credit for the other state, which mimics the income split). Your situation is different.
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