Here's another cover crop to try, if you think this will work for you.
Plant buckwheat. It sprouts quickly, and it shades out other plants (weeds), so that they don't get a good start. Let the buckwheat grow for a few weeks; let it flower if you want to feed the bees. Cut it down and either let it sit, or rake it away. If it's tall enough, the stems are long enough to wrap around the tines of a tiller, making it difficult to till in.
I did 2 courses of that one year, about 6-7 weeks each. Obviously, there wasn't enough time in a Michigan summer to grow anything, though I might have gotten some peas or kale or chard to produce.
There were almost NO weeds in that area the next spring.
Buckwheat does not fix nitrogen, and it will die completely with a frost.