Fresh water just needs to be boiled. The main problem in most rivers and lakes is just biologics and chemicals/toxins not so much. My last neighbor still had a well with a handpump. Almost never used so it had be primed first to get water. There’s rainwater collection too.
People WILL find a way to survive
Yes, they will. After about a year, MOST will be dead. Up to 90% of the country. That is a worst case scenario.
Of course if only 75%, 50% or even 10% of the population dies, it will be the largest casualty event since the Plague! And in terms of raw numbers, the absolute catastrophe the Earth has seen since WWII, maybe worse. 30 to 40 million dead is likely within the first month or two. It took almost a decade of war to kill than many in WWII. From the Japanese invasion of Manchuria to the clean up after the Atomic Bombs.
Imagine the permanent psychological damage a society would sustain if it lost 10% of their population.