72 hour visa free transit

Unfortunately, your segment:

Hong Kong - Chengdu - Hong Kong

Is just a return trip and not a transit. Both 72 and 144 hour visa free transit are eligible if a traveler is in a real transit. In other words, the basic rule of 72/144 hour visa free transit requires that the immediate points to your mainland Chinese stop have to be in two different countries or eligible territories (eligible territories are Hong Kong, Taiwan and Macau). As you can see you have one and the same place (Hong Kong) as your both immediate points.

My suggestion is that you try to adapt/alter your itinerary somewhat in order to return from Chengdu to Macau, a city located so close to Hong Kong. Like this your itinerary would be:

Hong Kong - Chengdu - Macau

or

Macau - Chengdu - Hong Kong

When you have returned to Macau, you will just proceed back to Hong Kong.So, if you can do that, you would be eligible for 144 hour visa free transit in wider Chengdu area and the visa would not be required.

Thank you for the information. I cannot change my itinerary so I will have to apply for the 10 yr Chinese visa. Can I mail in the visa application or do I have to go through a service since I am not near an embassy?

I apologize, but I am not familiar with the routine of embassy/visa offices located in US. The best thing would be to contact them directly and to see what their requirements are. I'm sorry.