For sake of clarity, those with extra frets should pretend your dulcimer does not have any + frets. They cloud the issue, though I have and use extra frets. Diatonic instruments and the music thoeory that they follow, do not go into extra frets.
D Aeolian scale is D,E,F,G,A,Bb,C,D. D Dorian scale is D,E,F,G,A,B,C,D. Note that Bb in Aeolian is replaced by B in Dorian. Many songs do not use all the notes the scale contains, and sometimes you can play these songs in more than one tuning. If you were playing chords, you would find that some chord notes are not available in both tunings. Playing melody and drones you have the choice in these songs.
However, if you play the same song in more than one tuning, but use the same fret numbers, you are not playing both versions in the same key. I suspect it is only the extra fret that allows you to do this, but it still doesn't allow you the correct drones for both keys. You may need to tune the drones to give you the correct drones for the key those frets gives you. The "standard" drones are the 1st and 5th notes of the scale. In the DAC/DAG example above, this is D&A. But if you use the same fret numbers in both tunings, you are not playing the same notes, and the drones do not match.