Hello, hopefully you've received some help beyond this post, because I haven't seen any replies here. LOL.
You don't necessarily have to play tritones in your left hand, but it does help to make your chords sound fuller. And yes, with each chord, your tritone will change.
Some organists use the left hand to play bass notes (if there's no bass player), while in other instances chords are used in the left hand and fingering in the right while bass notes are played by foot.
It all depends on your style and what works for you. I hope this helps a little.