It is very simple. Always use the passport of the country you are in if you have one from that country. Only switch passports in the air, never on land. Never show immigration officials you have two passports. It's never necessary and needlessly complicates your life and their lives. Only show your other passport to airline check-in staff if you need to show you have permission to visit that country.
If you mean show your US passport to airline check-in staff to show you don't need a US visa, then yes. Otherwise, no. You never show you have both passports to immigration officials. It's never necessary and needlessly complicates your life.
In general, the rapid antigen test is cheaper than the RT-PCR test. Both are accepted for entry into Phuket. It would stand to reason that it would be possible to get a rapid antigen test in Phang Nga (province to which Phuket is connected by a bridge) but I have not found anything online in English. A rapid antigen test can be found, for instance, in Bangkok for 1,500 baht:
Sometimes people are misinformed even if they are superintendents of a very large private school in Bangkok.
In my opinion, you are making something very simple complicated. Look at the graphic. Understand what it is saying. You don't seem to want to take yes for an answer. ;-)
I will try to make this as clear as I can as I appreciate there's been a lot of "news."
There are only two kinds of people in Phuket.
1) International Arrivals. What this means is that you arrive in Phuket at the International Airport on an international flight. In other words, you weren't in Thailand before and now you are and that arrival occurred in Phuket. You are subject to all the rules for international arrivals, commonly called the Phuket Sandbox Rules which will become official (and known) when the proposal is finally passed and published in the Royal Gazette, which is expected to happen in a few days.
2) Domestic Arrivals. What this means is that you are already in Thailand, just not yet in Phuket. You can arrive in Phuket on a domestic flight at the airport, or you can drive across the bridge into Phuket. The Phuket Sandbox rules (whatever they turn out to be) have absolutely nothing to do with you. Not one little bit.
Now, Phuket has rules for domestic arrivals. The rules are as specified in the following graphic.
Kathy, respectfully you are mashing together the rules for entering Phuket as a domestic arrival with the Phuket Sandbox rules for international arrivals.
For a domestic arrival, it couldn't be simpler. Just follow the rules published in the chart below. Nothing to do with the Phuket Sandbox. Look at the graphic. See the 1 with the words "Domestic Arrivals." See that there are three options there 01, 02, or 03. You pick one (1) and only one (1) of those options to satisfy the requirements to enter Phuket. Note: again you are a domestic arrival, not an international arrival. Don't become agitated about the international arrival rules (Phuket Sandbox) because they DON'T apply to you as a domestic arrival.
[As a sanity check, think about what it would mean if you had to satisfy all three (3) options. You would have to be fully vaccinated (or have one jab of AstraZeneca) AND you would have to have a negative covid test AND you would have had to have already had covid. That is not the case. It makes no sense. People are arriving in Phuket every day who have NEVER had covid.]
I'll grant the "news" has been buzzing with Phuket Sandbox "information," but if you are a domestic arrival it DOES NOT MATTER AT ALL. You need to follow the domestic Phuket entry rules ONLY. The Phuket Sandbox rules do not matter for the purpose of entering Phuket as a domestic arrival.
There seems to be a great deal of confusion on the issue of Phuket. There are two categories of people arriving in Phuket. International arrivals take place under the Phuket Sandbox proposal that is currently being worked out. Domestic arrivals don't have anything to do with the Phuket Sandbox. Nothing at all. The OP would be considered a domestic arrival. He would have to be vaccinated OR pass a covid test no older than seven days.