I am taking still and video photography for a film production. It is set inside a theatre that is filled with light fog. I somehow can't get sharp focus and because of the low light I can't up the shutter speed either making it double hard when shooting on the move. (The stage rotates, so I have to move). I am shooting with a 50mm prime and an 18-200 with an is. Any pro tips to get the photos sharp?
Is this a theatre piece that you are shooting for publicity or documentation?
If so, I'd talk to the lighting director and director/choreographer and explain the problem and see if you can arrange a dress rehearsal with more light. You can then either underexpose and/or post-process so the resulting images match the intended lighting effect.
There are some sets of conditions that are just extremely difficult to shoot in. Low light with moving scenery and people is such a set. My normal advice would be to bring your own light (e.g. an off camera flash) but with a stage full of fog that probably isn't going to work. In any case you are going to get some softness just from shooting through the fog. There's nothing you can do about this.
A monopod might help you reduce camera shake whilst staying mobile. Increasing camera sensitivity and using a faster shutter is the other option, but you will get more noise, you can try and remove it in software but my advice would be to just embrace the noise and do something with the colours (desaturation / black and white conversion) to create a certain mood for the image.
Sticking with the theme of "make it look intentional", you could go the other way and select a much slower shutter speed and move along with the stage/actors and get some artistic motion blur in there.