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Welcome to FT!
You can file a complaint with USPS Customer Service regarding incorrect information on the USPS website, as well as the difficulty in making an appointment by phone. If you do so, make sure you get a case number, just in case no one responds to your complaint within a few days.
As for the State website, I just did a Google search and found a page that said the appointments are necessary every day of the week at Ben Franklin Station. Perhaps the page was recently updated, or perhaps you accessed a different page. The State website also indicated that walk-in service is available five days a week at a non-postal location: the D.C. "Notary and Commissions Office"; would that be more convenient for you?