Answer: It doesn't sound like an insect problem, but a disease, so insecticides won't have any effect on the plants. Overcrowded plants with inadequate sunshine and air circulation can develop fungal diseases. Picking off the affected leaves will help stop the disease from spreading and will also open the plants up to better air circulations. As long as your plants are putting out healthy new growth, you're staying ahead of the problem. You might want to reconsider spraying your plants - fungicides are typically acceptable to organic gardeners and if it will stop the spread of the disease, it might be something you'll want to do after all.
Next year, be sure to plant your veggies in a different spot, just in case there's a soil-borne fungus at work.
Best wishes with your garden.
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