The reason it sounds off to our ears is because the drones are in D, while the melody is played in C ionian as Ken has it. You can tune to the "dorian" mode, but if you finger the ionian scale that's what you get. A point to remember is this: every mode is available in any tuning, by starting your scale on a different fret. But the drones must be tuned correctly to give you the harmony for that key.
If we start our scale on the open string, we get the mixolydian mode. Always. If that string is tuned to C, we have C mixolydian. To harmonize this, we tune the drones to C and G. If we leave that melody string tuned to C, but begin our scale on the 1st fret, we have the aeolian mode of the key of D. We must now tune our drones to D and A to harmonize this new key and mode. Leaving the melody string at C, and starting our scale at the 2nd fret, we now have the locrian mode of E, and we need to tune our drones to E and B. This continues all the way up the fret board, with each mode in a new key. Drone and chord players alike are bound by this. The mode not only gives us the notes we need for that modal melody, it also gives us the notes used by chord players to harmonize that melody. This is also true for drone players, just not as readily visible. But as we learn to hear, and to trust our ears, we recognize when the harmony is right for the key and mode of the melody. Playing alone, this is dissonant, but try adding a chord player to the mix. If you play this song tuned to DAG, what key do you tell your guitarist buddy to play it? In C or D? If he plays it in C, his chords will sound good against the melody, but will clash terribly with the key of D drones. If he plays it in D, his harmony and yours will work, but the melody will clash with both harmonies. The key of the melody string must match the key of the drones.