I share your pain, I have a heck of a frop of crab grass. I plan to do a second planting in a wide row so today I ran the garden tractor through the row and mowed. I used a cultivator with 3 potato plows mounted to uproot the frab grass. It will take a few more days of cultivating to kill the grass.
I wish I had laid large black plastic sheets down on the grass covering the entire area of the garden for a few weeks prior to plowing in the spring. It would have killed everything under it and my crabgrass problem would be insignificant.