The influx of patients is straining hospitals and pushing health care workers to the brink as deaths have risen to an average of more than 1,000 a day for the first time since March. Hospitalizations nationwide have increased by nearly 500 percent in the past two months, particularly across Southern states, where I.C.U. beds are filling up, a crisis fueled by some of the country’s lowest vaccination rates and widespread political opposition to public health measures like mask requirements.