Who is responsible for outstanding bills if a S Corp closes?

It seems you have multiple questions. As for who a bill collector calls, they are entitled to make phone calls in an effort to collect money. These rights are subject to many limitations. You can review the Fair Debt Collection Practices Act for an education on that subject.

As for who "is responsible" for the bills. That seems to be a two-part question. Who is legally obligated to pay the bills? That is typically only the corporation, unless someone else guaranteed the obligations. However, if shareholders took distributions of cash from the corporation, leaving the corporation unable to pay the bills, the shareholders are probably liable, to the extent of the cash received. However, regular wages earned do not have to be returned. If a corporation goes broke, it goes broke. Provided money was not wrongfully diverted to owners (or anyone else), the creditor is probably out of luck.

As for who is responsible among the owners to handle the financial affairs of a business that is near its death, the answer is . . . . nobody! While directors owe duties to the corporation and officers are required to perform services to justify their salaries, no one is required to remain a director or an officer. In other words, if it's a pain to serve in either capacity, a person can resign. However, those involved typically see an incentive to keep up with the affairs as doing so is likely to minimize the hassle directed towards the owners/former owners of the business.