I still use a camera mounted flash and a light triggered slave. The camera mounted flash can be set to bounce off the ceiling.
I have found problems trying to get shadows in the right direction, and have used standard LED lights, much depends on shutter speed at 1/30 second not noticed any problem with LED strobe effect. I think from memory with flash fastest shutter speed is 1/185 second, over that speed and I get cropping with standard flash, there are today flash guns that send out multi pulses so you can use a higher shutter speed.
I think my flash is around 1/10,000 second. Using LED bulbs designed to dim they flash around 100 times per second so a shutter speed of 1/90 second seems to work well, I think my camera can be set to 1/2000 second but that is not really the shutter speed, as the shutter in essence scans the image the first shutter is followed by second shutter so for both shutters to be fully open no faster than 1/185 second with my camera, some higher and some lower.
But fluorescent is a problem with colour, although tungsten is on the red side, as long as all lighting is tungsten it can be corrected, but because fluorescent produces bands of colour there is no way to correct, you can play with colour temperature slider as much as you want, but some colours are missing so you can never correct it.
What is more of a problem is different colour temperatures, take a picture in a room with two lights of different temperatures and again no amount of playing can correct it. The same when you get daylight and artificial light mixture, I know the collage where I studied photography the studio had no windows, so you never got a mixture of light, the flash guns provided the back ground light which was replaced with the flash. We had problems with the fluorescent lights, they caused odd effects from the coatings even when switched off, had to have diffusers covering the tubes so they would not effect the studio lights.