1 year old ongoing cold

Baby is 13 months to the day. We just relocated from KY to KS and he had a deep non-productive cough since before the move. Pediatrician said it's just a cold. After 4 weeks, took him back and they put him on Amoxicillin 20 day regimen for sinus infection. Ten days into antibiotic, cough is gone but broke out in rash so discontinued antibiotic and rash went away within a few days as we were told it would. So, now the cough is mostly gone, comes back at night a little but baby is so congested during night time and has thick yellow snot all day long! I look bad taking him to daycare, I'm sure but it's been weeks and I have to work! Can someone tell me what they might recommend? He still nurses at night but is having real trouble with the congestion. He doesn't seem to have much of an appetite at all.