Obviously, you have to shoot in combat zones, they are combat specific areas. But there are other non-combat specific things you can do in Odyssey, like salvaging missions, base powering missions, espionage missions and so on. All in all, Odyssey does add quite a bit of things to do.Can you do Combat Zones without shooting? Can hardly imagine that, however, you have now made me curious. I never did much of the on foot play yet (I'm out in the dark, but not quite under a stone) and all I know that it's basically one NPC team vs another NPC team, similar to the space CZs where you have to support your team in one way or the other - and if only by binding enemy forces which in principle doesn't necessarily require shooting.
And for what it's worth, I do like Odyssey's FPS model. When you get decent enough of framerate, it is fun. The big issue really is the tremendous framerate drops you get in conflict zones at the moment, which can get below 25 frames at the moment.