There are several different causes of strange odors on the breath. You are right that one of the most common causes of this is a tooth or gum infection. If you have been regularly brushing your teeth and if you are also up to date on your visits to the dentist
, then this is unlikely to be the problem.
Other common causes of this symptoms would be nasal allergies or chronic sinus congestion, which can lead to serious mucus collection in the nose and sinus, which drips down the back of the throat. This mucus can give off an odor which can cause bad breath.
Finally, another very common cause of strange odors and bad breath is severe heartburn, as the acid washes up into the back of the throat from the stomach and causes odor.
I would suggest that you start by going to see your primary care doctor
about the issue. They will be able to examine you and rule out these common medical problems (all of which are generally quite easy to treat). If this search turns up negative, they will be able to help you think of rarer and more serious causes of disturbances in your sense of smell, which seem unlikely at this time.