The problem essentially is that you blend/mix your (RGB, A) of the texture with (RGB, 1) of your fog color. This means that when the fogAmount is large (assume 1), then your grass quad essentially has alpha=1, so is fully opaque and thus you see the fog color that would only have been valid for opaque sections of the texture.
So, you must only blend the RGB channels between texture and fogColor, NOT the alpha channel.
Once you fix that, you'll bump into the next issue that your fog on rendered geometry is inconsistent with (missing) fog on the background. So, you will see grey fogged grass in front of perfectly clear background sky. This is of course physically implausible, since the light coming from the background (around the grass blades) is affected by the same fog as the light coming from the grass blades themselves (in fact, the background/sky light is affected by even more fog, since it travels a further distance than the light coming from the grass blades). So, given your current fog model, your background/sky should just be gray.