HELP!!! Organic Weed Prevention

I've had an organic garden for the past two years.  I don't by any means eat all-organic, but, I just can't bring myself to spray MY beautiful garden with chemicals.  Never really had a problem with bugs or weeds.  Hubby and I would trek to the garden about 4 weeks after our first planting, pull a few weeds, and be done with it once the intentional plantings got larger than the "grass".  This year is different!  Our garden now measures about 20x28 feet and has gotten a ridiculous amount of rain over the past few weeks.  The "grass" is quickly catching up with my tomatoes.  It is beginning to overtake the beans and carrots!

So, I tried the newspaper method.  It took an ENTIRE Kansas City Star just to surround three tomato plants and 2 dill plants.  I carefully soaked them down as directed and before Mr. Manda got home with the grass clippings he worked his butt off for all day (lawn care business), they had dried in the sun and blown up against my hog wire charming fence.  What's a girl to do?

Well, luckily unfortunately, my baby got in trouble at school today.  He only had ONE day left!!!  So, we have decided that he will be assisting Mr. Mama with the weed pulling tomorrow while Mama is toileting and dispensing pills to her 12 "guests" at the hospital.  Now, I can't count on my 13-year old son to get in trouble any more.  He really IS an angel.  So, I need a solution!!! 

Anyone???