Birdie, welcome to the forum! Please tell us where you are in the USA, and your growing zone--it would be so helpful in giving suggestions or advice.
Here in central Texas we have grasses of all sorts, and most of them grow best in our flower and vegetable gardens! Have you considered putting down cardboard between rows to smother the grass? We had an awful problem in our asparagus bed, and found that putting down cardboard and covering it with mulch (grass clippings because we are "frugal" and had lots of clippings from mowing the lawns) kept the grass within bounds.