My breath smell like mothballs, why is that? Could I have gingivitis or a sinus infection?

Yes and. Yes and could also be tooth decay, heavy plaque/biofilm build up, a leaking crown or bridge.
Bad breath. There are as many different smells of bad breath as there are reasons for bad breath. Specific bacteria called eubacterium, porphyromonas gingivalis, porphyromonas endodontalis, prevotella intermedia, tannerella forsythensis, and treponema denticola are the main culprits of odor-causing bacteria in the mouth. These bacteria digest proteins and Amino Acids that break down into different compounds.