I'm no fan of weed barrier fabrics. I have found they cause more problems than solutions. Preen, a commonly found pre-emergence that is easy to use, is what I like. Obviously you can't apply the pre-emergence until after the vegatables have germinated. For the rows in between the plantings, you might try newspapers with a light coat of mulch on top. That is a good weed barrier, that doesn't cause much future problems.