Clinch Mountain Backstep is played using a minor scale, which is the "Aeolian mode" on a dulcimer.
In order to play it in D-A-A tuning you need to put a capo at the first fret.
Here is a tab in D-A-A with capo at the first fret (which will actually be in the key of E minor):
The other way to play the tune in the correct mode is to change the tuning to D-A-G. Tune your melody string down one note.
Here is a tab in D-A-G Aeolian tuning (which gives you the key of D minor):
Of course, if you want to play the second tab in the "usual" key of E minor, just tune your dulcimer up to E-B-A.
Hope this helps!