Answer: I wish I could go out into your yard and look at your watermelon plant with you because I'm not convinced your plant is diseased. I've come to this conclusion based on your statement that the vines are growing fine. When a plant is diseased it usually slows or even stops growing. I don't know what the brown might be, but the yellowing could indicate a need for fertilizer. The oldest leaves on a watermelon plant can die back without anything being wrong with the plant. It's just that these leaves have served their purpose and the plant is putting its energy into producing new leaves towards the ends of the stems. If you're concerned about the older leaves you can simply cut them off the plant. I think this is a better approach than spraying with a fungicide. First, you may not be dealing with a disease and there's not much point in spraying if you don't need to. Second, removing the suspect leaves will also remove any disease pathogens from the garden. As long as the plant is producing new leaves and the stems are growing longer each day, there's nothing to worry about.
If you haven't already, try feeding your plants with a 5-10-10 fertilizer or side-dressing with some compost to provide nutrients to your plants. I think you'll see a difference in leaf color and in their growth rate.
Best wishes with your garden!
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