Cold or sinus infection?

I took Elizabeth to see Dr. Rowland again today. She got a cold back in February and it never really went away. Even after almost going through a whole bottle of saline drops and humidifying the nursery, these past couple of days her congestion has seemed worse than ever with a stuffy nose and coughing - as if a lung is going to come up any minute. If I touch her chest, I can feel the congestion! I spoke with Dr. Rowland's nurse on Monday and told her Elizabeth's symptoms. She was concerned about the congestion in the chest, so she suggested I come in. Dr. Rowland was out yesterday and I was unsure about having another doctor in the practice examine her. So I booked an appointment for first thing today. Dr. Rowland was concerned that she's been sick for 25 days now and thought it could be due to a sinus infection. After listening to Elizabeth's chest and looking up her nose, in her ears and down her throat, she said everything was clear - no fluid in her ears and no yellow or green mucus - so no symtoms of a sinus infection. Yet she still wrote a prescription for Amoxicillin. She said it won't hurt Elizabeth, but I'm concerned about giving my 3-month old antibiotics. She did say if things don't clear up within 4 days, I know it's not a sinus infection. I pray that my little one gets better soon!