I believe a thick layer of newspaper would help. I have a horrendous weed problem in my veggie patch as well (as well as other beds), but this bed was the only one that I room to experiment. Between 4 tomato plants, I put down a layer of newspaper, about 5 sheets thick, and covered here & there with some leftover bagged manure. The weeds were minimal, as where I didn't cover, and after hand pulling the visible weeds, another jungle had sprouted. Once my tomatoes & peppers are done for the season, I'll be putting down a thicker layer of newspaper topped off with a nice layer of shredded leaves, to hold down the newspaper. I'll do the newspaper thing again in the spring, for good measure ;)

It sounds like purple or yellow nut sedge. I honestly don't know of anyway to get rid of either of these weeds other than specific herbicides.

I've always pulled nutsedge, to get all the root or bulb, don't know any weed killer that's strong enough to kill the root/bulb without contaminating the soil.

Sifting might be effective, but fork the soil onto the sieve a little at a time;
if you till, you'll break off little pieces of the bulb & drive them deeper into the soil, & they'll grow.

During your wet season, early summer, before the plants mature pull as many as posible. It is time consuming but if you can pull the small plants they will not make more nuts for next year. The ground has to be moist or they will just break. About 5+ years ago I purchased bird seed that either had the nutlets or someone had a bumper crop of seeds. I decided to move the feeder and dig up the area for flowers when I started noticing these small bulbs. I sieved and pick as many as I could and put aside thinking they might be pretty. When I started seeing the plants come up into the fire the bulbs went. I keep pulling the plants as I see them and they are becoming fewer.