How do I prevent or reduce shake when observing with binoculars?
Don't hold the binoculars in your hands. Humans are made of meat. wobbly wobbly meat. There's apparently devices that let you mount binoculars on tripods - (this google search would be a start). Those and a tripod would probably be helpful in decreasing shake. 

I suppose it would affect mobility a little but that's a tradeoff.