Generally speaking, teeth do not cause sinus infections. However, if you have teeth that are so broken down that the inside layer of the tooth where the nerve is, is now exposed and resulting in an open pathway for the bacteria in your mouth to head into the bone area around your tooth, the resulting bone infection can then spread into your sinuses. It sounds like your main problem is to get the infection under control. A dentist can determine with you how best to control the teeth infection, whether by extractions, or decay removal and sealing the broken down teeth; a doctor can control the ear and sinus infections; together or individually they can treat the thrush. Once the infection is under control, then I would think that pain management will no longer be needed for your mouth and sinus conditions. Without dental insurance and with limitations in one's financial situation, sometimes a dental school is the way to get treatment. Best of luck in your difficult situation.
Dr.Regina S. Wong