... and the doctor says his lungs are clear, he's taken meds, and seem healthy anyway, then maybe it's time to do some spring cleaning in your home, especially the bedroom. Maybe it's time to change beddings, blankets and mattresses/pillows. Or at least, the sheets. You should also clean your electric fans and AC filter.
If all that doesn't get your child's sniffles improving, it could also be your laundry service's cleaning aids that's causing the irritation for your child.
And then of course, there's the diet. Feed him organic, natural, non-processed food. Some avoid dairy and other highly-allergenic food like nuts.
The point is, sometimes, a persistent cold and cough is not viral at all, but an allergy. Make sure you look into that before subjecting your child to numerous medicines he really didn't need.