Welcome to BYC. I would not do something as rash as culling my flock unless I got some testing done, or knew what was wrong. Bad odors from the beak can be from many causes such as sour crop, canker, and yes from the respiratory disease coryza. Coryza is pretty serious causing thick yellow nasal secretions, coughing, gurgling, and swollen gunky eyes with pus in the worst cases. He could be sneezing or chirping from dust in his feed, or the head shaking could be from something stuck in his crop, a respiratory disease, or gunk in his ears. So I would check his crop at bedtime to make sure there is food in it, and that it mostly empties by morning. If it doesn't empty or seems full squishy, and if he vomits brown fowl smelling liquid, he could have sour crop. Look inside his beak and throat for any yellow to gray patches that might be canker, a protozoan infection.
Sorry for your loss. Whole corn is not very good for chickens. Cracked corn or scratch grains with cracked corn are sometimes used as a treat (no more than 10% of the diet.) Layer crumble or pellets are best for 90% of the diet. The large kernels of whole corn may not be getting broken down in the crop. There also could be long grasses that are getting stuck and causing sour crop. Sour crop can be a result of impacted crop from decay of food or impacted grasses.I'm not very experienced with sour crop, but I would try to give as much water with electrolytes as possible, massaging the crop several times a day, to get the crop to empty. Do not feed whole grains while the crop has a problem, but stick to water, plain yogurt or buttermilk, soft or raw egg, and the second day if improved you can use some watery feed. Be careful if you vomit the chicken, since it can cause choking if not done properly. Here are some good links to read: