Can you still but a ticket at the airport?

I’m currently stranded at a very small regional airport that only serves Delta and United. My delta flight was cancelled, but there’s still one more United flight leaving tonight. I tried searching the app and calling, but the app claims it’s sold out and the phone line has a two hour queue.

I went out to the United check in counter and asked if the flight is full, they told me there are over 20 seats available but they “are not allowed to sell tickets”. They are telling me that airline employees can only check bags but they don’t have permission to sell tickets.

Is this correct? I can’t buy a ticket on a flight that’s not full at the airport anymore? I’m willing to pay whatever it costs, but it seems this flight will be leaving half empty without me on it. Can any United employees confirm this is true?