You don't seem to be talking about adding additional strings, just about changing the same six strings to different tunings. Two different things.
FWIW -- mostly we refer to tunings from the bass string to the melody, not melody to bass the way you write them... Tunings are usually written DAA, DAd, etc, where the lower case letters show a note an octave higher -- d versus D
Remember, the dulcimer is a 3 course instrument -- melody, middle drone, bass drone -- not a 6 course instrument like a guitar. Any course can have 1, 2 or even 3 strings, but we still retain the concept of melody, middle drone and bass drone strings. MOST dulcimer tunings involve the bass and middle drones being tuned a fifth apart, with the melody string(s) tuned to create different "scales" as the guitarists call them. D-A or C-G or G-D are fifths apart 1-5. Tuning the melody strings to different notes gives us scales-- 1-5-5. 1-5-7, 1-5-8, 1-5-4 etc. -- which start at different frets.
If you think strumming a 3-course dulcimer with 2 strings per course (total 6 strings) is "very cumbersome to get used to fretting..." and you're not getting "all that much fuller of a sound" -- them you're really in trouble if you try to strum a 6-course doubled string instrument like a 12 string guitar.
A 3-courses double strung (6 string dulcimer) tuned with some octave pairs and other combinations are not uncommon. Bass courses strung and tuned Dd are very common. So are melody course tuned Ad. Thus you have Dd-AA-Ad. Once in awhile you'll find someone experimenting with octave tuned middle drones Aa and you could have Dd- Aa-Ad. Any of these octave tuned couplets would need something other than the ordinary set of dulcimer string, of course.