On my last liveaboard trip, the guides were shooting videos with TG-4 cameras, and our guide was using the palm of his hand as WB reference - seemed to work out okay. However, the manual white balance on Sony cameras is limited to 9900K - last time I tried setting my A6300 to manual white balance underwater, it just kept giving me errors. On the other hand, I have read since then that despite throwing the error, it does actually set the white balance to an appropriate value, the camera just doesn't show it - I have another trip coming up in three weeks, will be able to test it then. In any case, the underwater white balance option (the fish sign, rather than auto - if your camera is running an early firmware version, you need an update to add it) seems to produce pretty good results.
Regarding focus - try shooting in manual mode, with aperture set to f/8, exposure to 1/60 (the '180 degree exposure' equivalent if you're doing 30fps), auto ISO, and focus set to AF-C with wide focus area. You should see little green squares dancing across your frame, indicating the areas that PDAF (phase-detection auto-focus) is locking on to. Narrower apertures give you greater depth of field, but f/8 is the smallest aperture that PDAF will work with - if you go to f/11 or higher, the camera is limited to the much slower CDAF (contrast-detection auto-focus) and the greater depth of field will be offset by slower tracking.
Regarding focus - try shooting in manual mode, with aperture set to f/8, exposure to 1/60 (the '180 degree exposure' equivalent if you're doing 30fps), auto ISO, and focus set to AF-C with wide focus area. You should see little green squares dancing across your frame, indicating the areas that PDAF (phase-detection auto-focus) is locking on to. Narrower apertures give you greater depth of field, but f/8 is the smallest aperture that PDAF will work with - if you go to f/11 or higher, the camera is limited to the much slower CDAF (contrast-detection auto-focus) and the greater depth of field will be offset by slower tracking.