Well, you've probably already got the LH "oom-pah" figured out, but that's where I started - in C, for instance, C-G with thumb/pinky on 1-2 (assuming you're in 2), and the RH chord around middle C on the off-beat.
Then I added bass runs in the LH - G-ABC- for instance, to get the V-I cadence. And a ton of variations on that to get from one chord to another - like playing runs on your guitar.
You can try splitting the chord in the RH, with the thumb on the beat along with the LH, and the other two notes on the off-beat. Or vice-versa.
You can syncopate the RH, playing it between the beat and the off-beat. This is really cool if you anticipate a chord change with it by playing the new chord on the off-beat just before the "oom."
You can use different rhythms in the LH - my current favorite is "tango" - dum, de-dum dum, dum-dum, de-dum dum, etc.
You can use 7 chords in either hand.
You can play arpeggio variations on the beat in the LH.
You can play "stop time" - just lay out for a measure or a phrase, or only play the first beat. Or you can hold a chord thru a whole measure.
You can move either/both hands to a higher octave for awhile.
You can do grace notes in the RH before the chord - little quick runs. Or glissandos like Jerry Lee.
I will still sound wooden when I'm working on a new kind of variation or rhythm, but once I'm comfortable or the piece gets inside me that goes away, and I suspect it will for you, too.
We don't have a bass so I think of my LH as the bass, but if you have a bass player the two of you will have to work out, mostly by playing together, how to dovetail your lines - but you're used to that just from playing in a band.
I love playing back up - have fun -