Most of my playing these days is a combination of finger-dancing the melody with my left hand and fingerpicking with my right hand, although I can, and do, strum chords to accompany my singing or to accompany another dulcimer playing the melody. I've tinkered with the chord-melody style, but I can't say I use it much. I prefer finger-dancing with my left hand or using a noter held in my left hand (thumb on top if the fretboard is high enough) for lining out the melody and fingerpicking or strumming with the right hand to get the strings to ring out and establish rhythm.
Since most of my repertoire is old ballads of Appalachia and the British Isles, I play mostly in 1-5-5 tuning (Ionian) and 1-5-7 tuning (Aeolian). The actual tuning varies with the instrument I'm using at the time. My 1-5-5 tunings include D-A-A, C-G-G, G-D-D, and A-E-E. My 1-5-7 tunings include D-A-C, C-G-Bb, and B-F#-A. Since most of my playing these days is done at home for my own satisfaction, the actual tuning is what I perceive as the best tuning for that particular instrument.
Although I have played in 1-5-8 (D-A-d) tuning on occasion, I seldom use it these days. I've found that the vast majority of melodies written for D-A-d tuning (which is a traditional Mixolydian mode tuning) are actually Ionian melodies more suited to 1-5-5 tunings, such as D-A-A. Although D-A-d tuning is the predominant tuning these days, I find it more suited to the chord-melody style. Since I prefer to play in a melody-drone style, the various 1-5-5 tunings are more useful to me.
The tuning and style you choose to use is a matter of personal preference. There are those who swear by the chord-melody style. I happen to prefer a melody-drone style. All styles are valid, and I make no value judgements concerning which style is better. The only thing that really matters is which style brings you the most satisfaction.