To "shower" a cham, put either a live bushy plant or a big fake plant in your shower, aim the spray so it bounces off a wall onto the plant (less force and the cham can move in or out of the path of the spray), and turn the water on luke warm. Remember, water that feels warm to YOU will feel hot to a reptile. Leave him in there for 20 minutes at the least. More is better if you can. The very high humidity, gentle warmth, and constant moisture will help rehydrate him and help loosen the discharge in his eyes. He may not like it at first, but most chams eventually love their showers. Do this every day or so if you can.
You can help clean the matter off his eyelids by squirting the saline directly on the eyelid and VERY gently wiping away any matter with a Q tip. What I'd suggest is first clean the eyes, then put in the drops. You have to handle him anyway so its a good time to do both. The drops won't hurt anything, so continue with them. Never stop an antibiotic treatment early or any resistant bacteria still present will just develop again and be harder to treat.
So how long has it been since he's eaten? If I remember right, its only been a few days. Is he drinking at all? You can give him Pedialyte in a dropper if you can get him swallowing, and this will give him electrolytes and glucose. Again, keeping him drinking is more important than eating right now, and force feeding is very stressful and risky. Forcing water or food can damage the esophagus and trachea.
He's probably uncomfortable, but a cham is not necessarily dying if it is reactive to anything around it and is attempting to drink. I wouldn't say he's close to death at this point...I have rescued a female jax who was in much worse shape and she eventually did very well. A cham who is close to death will not be able to perch, swallow, or control its tongue, legs or tail, will be completely unaware of its surroundings, eyes will be completely sunken in, and show odd coloration in large areas on its skin...such as one whole leg, its head but not the body, one side a different color than the other (not basking).
But, if you really feel he's far gone you will have to decide for yourself when to put him down. Let the vet do it by injection. There are other "home" methods, but whether they are as humane or not is really a debate.