It does look like early blight, which most of the time comes later in the season. I concur about the plant spacing. You need to have at least three feet between plants each direction. Air flow around the base is essential. It helps reduce conditions conducive to disease.
Mulching is a great idea and something is better than nothing. I have tried plastic mulch but went away from it a long time ago. Water will not pass through to the plant base. My mulch of choice is layers of newspaper and 6-8 inches of weed free straw. Keeps weeds down, watering at the base is possible, keeps the plant root zone cooler...all the things mulch is supposed to do.
Treating blights at your stage is difficult. Besides the peroxide, there are fungicides on the market that will reduce the spread and prevent more disease but not many cures out there. Prevention is key. Proper air flow is major. Also proper spacing so each plant receives good nutrient uptake. Crowded plants are stressed and are unable to fight off diseases. Do not overhead water and don't water in the evening or night so that the leaves are wet overnight. Fungus playground. Good sanitation, particularly in the off growing season is essential to keep the funguses and other diseases at bay. And a proper mulching program keeping the leaves and the soil apart helps since most blights and other foliage diseases are soil borne.
It is amazing, but by thinning the numbers of tomato plants you will not generally lose production. Healthy, non-stressed plants will produce more. What are the varieties you are growing? Some do better than others at fighting off disease.
Some ideas that may work for this year but something to think about.
