Here's what I'd do at this point. If not already dead and mowed, cut the field again and over spray with a 2% by volumn , 40% glysophate (RU)-water mix at a rate of about 15 gal./acre. Be sure to add an ag water softener to the water before mixing the RU. Hard water reduces the effectiveness of RU. Generic RU is good enough .... Eraser, Big-n-Tuff, etc. .... and is a big savings as opposed to Round Up. It is the same stuff chemically as far as killing grass !! . And for sure a 2% mix will kill the grasses big time. If you have broadleaf weeds in the plot too such as nettles, vines, or sapplings, etc. you may want to consider adding 2-4-D to burn down the broadleaf stuff. Probably too late for this now because this product works best in hot weather and needs to have 3-4 days of no rain.
Wait 5 days or so and disc or till the plot thoroughly. Save yourself 50% or more on seed by avoiding any of these "commercial" mixes such as Shotplot. Go to any decent farm seed/feed store and buy purple top turnip and/or any other member of the brassica group by the pound ... and make your own mix. Broadcast over your plot at a rate of about 3#/acre. We use a hand operated spin spreadeer. (You can save in bulk. Any left over seeds will be good to go next year if stored in a cool dry place, such as a fridge.)
Drag in the seeds lightly .... preferably use a cultipacker rather than a drag. These small seeds need not be buried if cultipacked. And if buried, only very, very shallow. If it stays dry, no big deal. These seeds will sit and wait for wet conditions.
The group I hunt with buys one 50# bag of purple top turnip and use these to overseed our plots (35 acres worth) which will have already been planted with a mix of wheat. oats, grain rye and Austrian winter peas. We make one pass with the cultipacker and wait.
You should still have plenty of time for the food plot to develop unless your season starts earlier than down this way .... which is Oct. 15.