To answer your question, I ran a hypothetical "John Doe" tax return through the TurboTax Minnesota state module to see if I could replicate what you're seeing. (My screen-capture images are posted below this text for reference; simply click to open.)
I ran into the screen that you appear to be referring to, and I tried answering the question both ways, in fact, to see what would happen on my Minnesota tax forms. As it turns out, it made absolutely no dollar difference to "John Doe's" state tax refund what I did.
In fact, the only difference I noticed at all was the appearance of an "X" on the first page of the main Minnesota M1 tax return; which is an indication to the state that this is a new mailing address for the taxpayer (in your instance you can safely ignore the mention of a first-time filer, because your move was in-state, and you've already filed with Minnesota in the past).
As such, given what you describe of your circumstances, I would respectfully suggest that you answer "yes" to the question on that page -- as it pertains to your new mailing address. That "flag" (or X) placed on page 1 of Form M1 will cause the Minnesota Department of Revenue to update their computer address records accordingly, to reflect your current information.
Thank you for asking this question.
