Hello and hope you are doing well.
Its unlikely that foul smelling stools are causing the constant bad smell in the nose. Your symptoms could be due to olfactory hallucinations. Phantosmia most often occurs as a result of a head injury or upper respiratory infection. It can also be caused by temporal lobe seizures, sinusitis, brain tumors, migraine, Parkinson's disease and stroke. You may need proper evaluation by your GP, please consult him for a neurological assessment. He may then refer you to a neurologist if needed.
Hope this helped and do keep us posted.