Howdy. I'm sorry that your chickens got flooded. Weather and farming don't always work together.
We found that shavings in our outdoor yards got matted and nasty. You will probably need to turn the sawdust to keep it loose. It will also likely get trampled down into the mud.
Not a criticism, just some notes based on our experience.
My wife found a local place that will deliver small amounts of wood chips. I have a 20 cm thick layer in our poultry yards and it's the best thing ever.
The sawdust will get you through but I would plan to switch and try to find a supply of chips before the snow flies if you can. Some local tree services will deliver part loads of chips.