Since the unanswered questions would vary from one respondent to another, you might end up having all questions in your export.
Instead, you can create a layout of required columns and export data in this layout:
https://www.qualtrics.com/support/survey-platform/data-and-analysis-module/data/recorded-responses/#SaveLayouts

Thanks for the response. When I go to Tools | Choose Columns I don't see any of my questions. I fear I just have too many questions in my survey.

I don't think it matters how many questions you have. You can still choose columns and save layouts to export data.