Personally, I think it's a mistake to think in terms of chords for traditional music. Alternative tunings like DADGAD are great for getting away from chords. Instead play a melody with a drone (not necessarily the same one throughout), a melody with a bass line, a melody with a countermelody (plus maybe a bass line also). If you sing, have the guitar play in unison with the vocal line with a counter melody above or below it and a bass line underneath, or just miss out the melody altogether and play the rest. So much more interesting than chords.

The reason everything sounds the same so often in DADGAD is because too many players only use it for the key of D and, if they want to play in another key, they use a capo and so get just the same sound at a higher pitch! In DADGAD you should be able to play easily in the major or minor of D, C, F, G and A: different key, different sound.

Happy exploring.