@floralin, there are a couple of ways to interpret your question here, so forgive us if we offer answers that aren't exactly what you are looking for. Plenty of people play only in a drone style on the dulcimer. They never play chords but always allow the bass and middle strings to drone while they play the melody on the (you guessed it!) melody string.
Other people play chords but mix in drone stuff. Stephen Seifert is probably the best player in theis vein. You might hear a rendition of a tune in which he plays it drone style first, then he adds some "partial" chords so that there is always one open string, then he might do a verse truly chord melody in which every melody note is accompanied by a three-finger chord, and so forth. I sometimes play drone style in the way you are describing. For example, on the B part of Soldier's Joy, especially if playing up the fretboard, I often just play in a drone style because I find it too hard to play chords up there up to speed. But when I play across all the strings in the lower octave, I do indeed play chords because they are easier to reach. So my rendition of that song uses chords some of the time and drones some of the time.
And just because someone is playing with open strings does not mean those strings are drones. It is often possible to play full chords using open strings. If you are tuned DAA or DAd, any time the chord is a D you can play all open strings, yet those are not drones; they are chord tones.
And as @strumelia explains, many of us do not always strum all the strings all the time. In any rendition of a song, I sometimes pluck single strings, sometimes play two strings, and sometimes play three strings (which may or may note be open strings). In a lot of my arrangements I play a chord on the first beat of a measure but then I just play melody notes until the next measure or the next chord change, whichever comes first.
And some players play in a flatpicking guitar style in which they rarely play more than one string at a time. Check out some of Larry Conger's playing for that style. They are playing chords, though, but as arpeggios or broken chords rather than block chords.
As @ken-hulme says, there are no right or wrong ways to play this instrument. There are many possibilities out there and you should play in the style or styles which speak to you most saliently. If it sounds good to you, do it. If it doesn't, then try something else.
Dusty T., Northern California
As a musician, you have to keep one foot back in the past and one foot forward into the future.
-- Dizzy Gillespie
updated by @dusty-turtle: 06/24/20 06:15:35PM