Oats are a LOT cheaper for a 50 lb bag than grass seed. Dunno what the weather is doing at your place. In September, I planted oats a couple of years ago in my big garden, ~35 x 12 ft, and I am further south than you.
You Could plant now and cover with a tarp when you are getting frosted. Might work.
Here, oats are $12/50 lbs
I would suggest that you mow and add grass clippings and leaves and used your spade and rake to work them in.
You could also cover your garden with cardboard and put down bricks or wood or weights. When the soil is "buried" the worms and microbes and fungi will be able to work better. Use your hose and wet down the cardboard.
EVERYBODY and his brother and DOG has extra cardboard, or get it from grocery/box stores. They stock 1x/week and have to dispose of it.
Also, call city dumps, who COLLECT it from city pickups.
Still you live in a HUGE rural state, you may have to do some legwork.
If you have recently moved, there is something fun you can do with all those cardboard boxes besides fill up your recycle bin. Reusing cardboard for the garden provides compostable material, kills pesky weeds and help you get a new bed ready quickly. Learn more here.
www.gardeningknowhow.com