Harlan - In addition to all the great advice above, you might also consider experimenting with alternate tunings. I use ADAE mostly (on both OM and mando), and LOTS of seisiun players use GDAD on zouks or OM's. This gives you lots of great open/droning options.
I play OM most of the time in my small group, and hold the mid-range/lower register. I'd call what I play rhythm sometimes, but it is rarely just strumming chords... I do a lot of moving lines, snippets of harmonies against drones, played rhythmically to accentuate what's going on makes it rhythm... not just the block-o-sound full chords. Sometimes crosspicking. etc. And sometimes when I feel like playing mando instead of OM, I do the same on mando.
Otter OM #1
Brian Dean OM #32
Old Wave Mandola #372
Phoenix Neoclassical #256
If you're gonna walk on thin ice, you might as well dance!