J.S.
Why are sinus infection and viral in quotes? Do you think that those are incorrect diagnoses? If your son has viral infections then the antibiotics will not work. That would be likely why the other two antibiotics didn't work, and a third wouldn't either. Your doctor was hesitant because there's a high chance this third antibiotic won't work and all it would do is build up resistance to the antibiotics for when he needs them for bacterial infections.
When there's a viral sinus infection, you can still treat the symptoms with over-the-counter treatments such as Tylenol for aches and pain and fever. You can also treat his sinuses with saline solution to help break up mucus and relieve his sinuses of pain and pressure.
The Tylenol would also help ease the ache from the flu shot. The shot should have been given in a soft, fleshy area and not a muscle. Rub the area gently to help distribute the medicine. A nice warm bath will help with the aching and soreness too. But yes, it's normal.
A runny diaper is likely NOT from the vaccine but could be from swallowing a lot of post nasal drip from his sinus infection or is a sign of an intestinal virus. Give him lots of fluids like soup and whatever he'll eat that's nice and healthy. My girls like fruity teas and non-caffeinated herbal teas when they're sick. My middle daughter loves ginger and other spices, so she also loves vanilla chai tea when it's hot. It helps soothe the throat and get fluids in.
Personally I would hold off on the antibiotics and let the other ones run their course. They're still in his system for a while anyway. Wait and see. His body knows what to do.