How can I swap my O-A visa to a new passport at the Thai Royal Consulate in Sydney?

You cannot move a visa from one passport to another. If you are on an extension, you can have that moved, but not a visa.

If you are actually still on the original visa full page sticker that was issued by the embassy, then you will need to carry both of your passports with you until the visa runs out. They switched systems to e-visa in March, so at most you'll need to carry both passports for 6 more months.

E-visa would be the same, but now you'll have to carry 3 things with you. Both passports and the e-visa printout, and hand all 3 of those over when you enter the country.