Kriko, the trichordo has its own techniques, as expected of course.
Modern "bouzoukists" (they hate to be called bouzouktsides) tend more
and more to play as with a tetrachordo, i.e. spreading the ambitus of
the melody over all three strings, in this way avoiding changing
position of the left hand frequently and quickly, to cope with the
evolving melody.
The older style uses drone, as you mention. But here again, there are
older and newer techniques. The classical bouzouktsides of the 50s
and 60s would play the melody mainly on the two upper strings, often
using diplopennia, thereby leaving the bourghana (lower D) for drone
and only occasionally for melody, if the ambitus goes too broad on the
low pitch end. The drone comes either on open D, if it matches, or by
pressing on selected frets, depending on the chord applied
momentarily.
There is also a much older style, that derives from the tamboura of
older times: for the bourghana the thumb is used, reaching the string
from the upper side of the neck, of course, while the four other
fingers are reserved for the melody. This technique needs lots of
practice to work efficiently. But you will see hardly anyone today
who still plays this old way.
When playing in a pitch other than D, a technique similar to the
Barree of a guitar can often be used: applying a combination of the
two first fingers on one selected fret and picking on all three
strings simultaneously, an effect comes up that resembles striking all
three open strings together: a fourth and a fifth simultaneously, but
in a chord other than DAD of course. For example, a "barree" on the
second fret would give an E major chord, without the third of course
(i.e. one B and two Es).
The way you describe, using the two lower strings for drone, is not
recommended: having to play the melody one one string only will never
give you the versatility that two strings offer, especially on quick
passages.