Usually having a bad odour is due to some problems excreting some toxins. Whatever it is, it could mean some metabolic disorder that can affect the way the mind works and cause emotional disorders.
I am glad to hear that you are trying to avoid psychotropic medication, as these only mask symptoms and do not "cure" the underlying biochemical disorder that is responsible for the symptoms. Most "mental" problems are due to some illness in the body - preventing the production of feel good neurotransmitters - AND NOT DUE TO A ILLNESS OF THE MIND!
As a psychonutritional therapist always start with investigating the most common causes of mood disorders and then move on to the more complicated cases. Sometime we need the help of nutritional doctors who are trained to test for metabolic disorder that can be treated nutritionally and without recourse to drugs.
The most common cause of mood disorders is Hypoglycemia, that can trigger the release of stress hormones. The secretion of excess stress hormones such as adrenaline lie at the root of most symptoms of mood disorders. Excess adrenaline production is also responsible for excess sweating and this may be implicated in "smelling" or bad odours. I suggest that you read the following articles:
The most common silent disease resulting in depression is hypoglycemia. Most people feel better if they adopt the Hypoglycemic Diet. However if problems persist I suggest you ask your doctor to be referred to a Nutritional Doctor, Clinical Nutritionist or a Nutritional Psychotherapist, for further tests and diagnosis and treatment.