I (IMVHO) would take him back to the vet and insist on an xray or two. Puppies this young and traumatized need to be checked for pneumonia (if his nasal discharge is thickened or colored/cloudy) and at the same time he can check the liver for size (relative to body) and for congenital defects (AAI and Chiari). If his liver appears small for his size, no vaccines until Liver Shunt can be ruled out (at 16 weeks a Bile Acid Test can be done). And no vaccines until congestion is cleared up.
It is
extremely important to give the meds as prescribed by your vet and to
finish all the meds. Failure to do so can cause bacteria to become immune to the medicine, then the meds will not help him, and he will continue to be sick. So please finish the medicines and give them to your pup as directed.