So I hung out in Hong Kong for one more night, which wasn't so bad, and the next morning I headed to the US Embassy.
Ironically, no one working at the door of the US Embassy speaks English. But, it all worked out. One of the officers working in the front let me use his personal cell phone to call the US Embassy Hotline so I could find out how to get a Vietnamese Visa. That person told me that I had to go to the Vietnamese Embassy. Once I was at the correct Embassy, the sweet people of Vietnam issued me a visa in 15 minutes and I was allowed to fly out and spend the next 30 days there.