Ok. The situation is not at all what I expected, but at least some of the solution is easy. A sufficiently experienced photographer can use mixed light sources by carefully adjusting the colours with filters on the light sources, but I've never known any who would bother to try it by choice. So for all lighting, stick with one type of light. In this case, if your light boxes have standard light bulb sockets, then just "retire" the fluorescents and replace them with the same daylight bulbs you are already using. Later you can replace them with LED panels which can be colour adjusted. You will be able to use adjustable LED panels with the "daylight" coloured bulbs or with normal "tungsten" coloured bulbs. Many of them are adjustable for power. If you have to chose, get a panel that is colour adjustable even if it is not power adjustable.
[2019-02-06 17:14 re-written for clarity.]
For the saturation, that will probably be adjustable in that camera -- I don't use camera, so I might be wrong about that. You should be able to reduce saturation in your editing software's colour adjustment. So that should be simple enough to do in post production.
[2019-02-06 17:14 re-written for clarity.]
About that last image. What colour is your backdrop? If it is grey, then the whole image has shifted to reduce the green from the fluorescents.
In general, there is no difference between still and video lighting theory, so there is a lot of information "out there". Search for "color balance". Here's a link that might help:
"The A to Z of Photography: White Balance",
By Rod Lawton July 27, 2018, Tech Radar
"
The A to Z of Photography: White Balance | TechRadar"
[2019-02-06 17:20 re-written for clarity and added link.]
The only general difference is that video makers tend to prefer "flatter" lighting because it allows the "talent" -- the people in front of the camera, to move around more freely. I think it was first done in the 1950's for "I Love Lucy", and is still commonly used for most TV shows.