Lack of oxygen won't be a concern, we got enough for millions of years. Sure it will run out eventually but not for a long time. So yes eventually it would kill Earth's fauna but not before evolution could make a dent in it.
Food supply will indeed be a much larger issue. Within days to weeks the majority of herbivores will start to die from malnurishment. Remember that it's generally the carnivores that do well without eating regularly. Plant matter is much harder to digest and generally requires a much larger portion of the day directly dedicated to eating.
So with a large portion of the herbivores dying there will be a surge in available food for carnivores. But this will quickly rot and then starvation will set in for our predators as well.
Within months the majority of species will have simply died off. The rest will likely quickly follow. Seas will become cesspools of disease and toxins. Humanity will be among those that die out. We can't survive on a diet of pure meat. Long term storage of food will remain for a long time if limited to a small group of consumers.
There are a couple ecosystems that don't use photosynthesis to supply all their energy. These get their energy from heat-loving bacteria that live on underwater volcanoes and get their energy there. These ecosystems are very deep in the ocean and include worms and blind crabs.
What humanity will do, there is no objective answer here. Will we die out? I see it likely if we can't regrow any plants ever.