Congested coughing baby

My son will be 4 months on Sunday. He’s had a mucousy cough since November. The doctor we saw said his lungs were clear and his oxygen was 100%. Nothing changed until the day after Christmas, we both woke up sick with a cold, he’s very congested in his nose and chest, I’ve been sucking his boogers out with the nose frida, he’s been coughing worse than he was before and we decided to call the dr today. We had a zoom call with his regular doctor and his doctor immediately said it was his formula and a milk allergy. He said he wants him switched to soy. It’s his doctors last day and next week we will be seeing a new doctor for his 4 month physical and I do plan on talking to that doctor before I switch anything. His doctor did this once before as well when my son was about 6 weeks old, said he needed to be switched to Similac Alimentum and my son had a horrible reaction to it and I immediately switched him to Similac Sensitive and the problem we were having was resolved. Does this sound like a milk allergy? Or does he sound sick? Because I think he’s sick and I don’t think the formula is the issue, but I really just want some other opinions.