Experts and BHG readers answer.
The best solution is to spot treat each weed with herbicide. You may use a sprayer as long as you use some kind of shield to protect the surrounding seedlings from drift. The simpler application method would be a "Weed Wand" that allows you to dab the herbicide right on the weeds. Weed Wand is a name brand for a long narrow tube with a sponge dab applicator on one end. If you attempt to pull the weeds by hand, you run the risk of disturbing/uprooting the tender seedlings nearby. The good news is that the initial major work in establishing a planting from seeds like this will pass. You're feeling overwhelmed by the weeds now, but keep at them. By next year, the problem will be a tiny fraction of what you're dealing with this year. The year after that your plants will be well established and the weed problem will be very light duty.