With help from other community members and with a lot of time, I can report: Problem resolved!
I navigated to the 1099-DIV panel and clicked edit. I did not change anything but that action allowed me to navigate to the panel where you enter the amounts for your state and for other states. I deleted both entries (i.e. the NC and Multiple States $s). I had to select the radio button designating all tax-exempt income being from one state and selected "Multiple States" as being that "one state". When I did that, the NC Refund amount at the top finally reset to represent all of these $s being from outside of NC (previously it treated those $s as all being from NC regardless of my actions).
I navigated away from and then back to the same panel. This time I re-selected the radio button for dividends from more than one state and re-entered the same amounts that I had previously deleted. The NC refund amount updated again, this time FINALLY to the correct value.
It shouldn't be this hard. If I had not be aware of what my NC refund should be, I might not have noticed TurboTax's error. TurboTax set me up to file an incorrect state tax return.
