But you didn't answer my question.
Because changing the shutter speed does directly affect exposure.
Is this not the case with the tg6?
If the main light on the subject (i.e., typical macro) is the strobe, the shutter speed is irrelevant unless it is so slow that the
ambient light overwhelms the strobe illumination. Shutter speed (if you could manually vary it on a TG-6) will affect the
ambient light exposure. For example, if you could choose 1/500 instead of 1/60, the illumination from the strobe would not be affected, but the
ambient light exposure would be greatly diminished, which is useful for making the water background black or deep blue instead of light blue or almost white.
For simplicity, if you are shooting macro with a TG6, you want to make sure you are using the underwater macro mode (the one with the fish icon). If you use P or A or Underwater Snapshot then the camera adjusts for the ambient light it sees, and adding a strobe will overexpose the picture. You can override the automatic exposure by forcing the camera to underexpose with -2.0 exposure compensation and the smallest possible aperture and lowest ISO. That is, for example, the Backscatter recommendation at
https://www.backscatter.com/images/.../Olympus_TG-6_Settings_Guide_Macro_Strobe.pdf. If your strobe has TTL, use it. If not, shoot, evaluate, adjust the strobe power, reshoot.