COVID-19 resources. Oswego Health. 110 W. 6th St. Oswego, NY 13126. 315.349.5511. Map and driving directions. If you are experiencing symptoms of cough, fever or shortness of breath, please call your primary care provider or the Oswego County COVID-19 Hotline at 315.349.3330. The severity of coronavirus symptoms can range from very mild to severe..