Help with cover crops

So we just bought 5 acres of clay in southern Ohio. This field has been used for corn and soy, and has been already sprayed with roundup for the season. I want to plant some cover crops now, and I need help figuring out What kind of cover crops are best for this land (see picture below), having in mind we are at the end if May, and whether to plow before planting or just spread seeds on the surface. Any advice is much appreciated!  

Do a search for Gabe Brown's cover crop seed mix, he does 14+ different seeds.  On 5 acres, you don't need more than one bag of anything but maybe oats or wheat.  You can buy bird seed and feed oats from the farm supply, and deer plot mix for clover and chicory, and grocery store bulk.  Anywhere you can get them, cost isn't a big driving factor as much as just get the seed in the right quantity.

Don't till.  Maybe a little disking if you have to to incorporate the seed, but any tillage reduces organic matter and releases carbon to the atmosphere.

I have seeded 30 acres with a riding lawn mower, pull behind spreader, and a scrap of cattle panel and old tires as a drag.  I have also over seeded the same field walking with a bag of seed and a $10 handheld Scotts spreader.

R Scott wrote:Do a search for Gabe Brown's cover crop seed mix, he does 14+ different seeds.

Thank you for the lead on the cover crop supplier! He even has phacelia, one of my favorite cover crops. I just placed an order. Hopefully it will arrive soon.