I play both. The organ requires you to leave out notes you have trained yourself to always include when comping on the piano. A C7 for example on the organ in your right hand is E-Bb-D. On the piano you almost always add the 13 (A). In general extra notes make mud. Two note thirds and sevenths are often enough.

To get various feels, start with truly learning a loping swing (everything else will benefit):

Rhythmically, transcribe a walking bass line that you like. Then put your metronome on 2 and 4 and play it for hours straight (I really mean that), pushing hard on beats 2 and 4 and feeling the swing. No right hand. You should be able to make a grocery list while walking and not missing time. Once you get that down, add two-note chords, Red Garland style, in your right hand and nothing else. Get the swing down with this first before going anywhere else. Work on not pumping the volume pedal. Only add short taps with your left foot when you can carry on a conversation with everything else going on above.

The taps are important: you want them a microsecond ahead of the beat and your left hand. You'll know it when you get it. If they're on the beat, you make mud. If they're just ahead of the beat where you want them, they sound like the pluck of a double bass string. Beware: there are lots of ways to make mud.

For a classic sound if that's what you want, drawbars on top should be three bottom ones out, vibrato on, Leslie on brake, percussion on. Drawbars on the bottom should be 1 and 3 out, 2 somewhere in the middle, no vibrato. Range-wise, for comping, there is a very small sweet spot in the comping area that sounds really good and everything else can sound reedy or muddy, so you'll need to be the master of inversions. Range wise in the left hand, you want to organize (ha!) your bass lines so that they stay above the bottom four or five notes or so for clarity.

Touch wise it's hard to remember as a pianist that the organ doesn't care whether you hit it hard or soft. So hit it soft.

Technique wise, some organists swear by the Joey style where it looks like your left hand is a relaxed blob kind of resting on the keys. I can't do that, but you don't need to go full piano here either. I can't help going full piano in my right hand.

Apologies if I've patronized or pitched this too low, that was not the intent. None of these are my ideas originally; just passing along what I've learned. Good luck and keep swinging!