What are the steps to transfer my visa to a new passport while in Thailand?

Yep did mine last year .. took both books to immigration and they transferred everything . I’ll pm you a pic of what it looked like after 😉

The Australian embassy don't give a separate letter, they just put a Stamp in the old passport with details of the new one issued in its place.

Once you get your new passport (AND the letter from your consulate stating you replaced a passport inside the country) you take them both to the immigration office to get the stamps moved.

They will not move the O-A visa sticker. instead they'll make a notation in the stamp transfer that you previously held an O-A in another passport and entered on it xxx date,

You can move the stamps and apply for a new yearly extension of stay at the same time

TM30 is fine and you won't do anything until you need to file a new one (if you ever do)

In Bangkok there is a "transfer stamp" form you fill out. Here's the back of it telling you what you need (again only in Bangkok).