Nate, as Ken and Skip have explained, the notion of key on a modal instrument gets a little tricky.
I would just define a key as the tonal center of a piece of music, the tone that seems to represent rest or resolution with the other notes creating different degrees of tension.
Your effort to determine key by examining the sharps and flats of a piece makes sense in western classical music, and you are correct that the key of D major has a C# and they key of G has a C natural. That "key signature" defines the major scale, or the Ionian mode. However, with traditional, modal music, any mode can be played in any key, so the key, or tonal center, does not necessarily determine the scale pattern. To use the most common examples, D Ionian uses the C sharp, but D Mixolydian uses the C natural.
In fact, those examples explain why the 6+ fret was added. On a true diatonic dulcimer tuned to D, tuning DAd would not give you a major scale (Ionian mode) precisely because the 6 fret is a C natural. To play the major scale, one would tune DAA and start the scale at the 3rd fret. Then you get the C# on the 9th fret. To avoid having to retune, dulcimer players about a half century ago began adding the 6+ fret so that they could play in the two most common major-sounding modes, the Ionian and Mixolydian, without re-tuning.
Let's also remember that a lot of folk and pop music doesn't use all the notes of the scale or mode. A lot of music is pentatonic, meaning only 5 notes are used. And heck, the old song by the Chrystals, "Da Doo Ron Ron," only has three notes in it!
"So what?" you might ask. Good question. My point is that every song has a key, meaning the tonal center or "home base" even if it does not make use of the scale indicated by the key signature.
Having said all this, I would guess that 90 percent of the time when you are tuned DAA or DAd you are playing in the key of D (or Bm, the "relative minor"). If you fret across all the strings, then you can also play in G and perhaps (though it gets tricky) A. I've recently been arranging several tunes that work in both D and G on the DAd dulcimer. That way you can modulate after a couple of verses and impress your friends and family.
Dusty T., Northern California
As a musician, you have to keep one foot back in the past and one foot forward into the future.
-- Dizzy Gillespie
updated by @dusty-turtle: 10/05/20 01:30:02AM