The Australian Embassy in BKK no longer issues letters confirming your Thai address and Bangkok Immigration won't do it unless your 90 day reporting is up to date.
For your Thai DL, there's a grace period of 12 months to renew an expired license, so if you still have a week or 2 window to renew before the 365 days lapses, then you can use your Yellow Tabien Baan for confirmation of your address at most DLT's, some branches do not accept them but I know from my own experience that both DLT Bangchak accepts the yellow book. Bear in mind that there is a 3 month waiting list for foreigners at Bangchak, so looks like you're going to have to get to DLT Chatuchak very early in order to get it done.
Similarly to yourself my Thai DL had expired in Jan 2021, and due to lock-downs and closure of DLT's, I was faced with the prospect of being unable to renew by this month, given that the DLT Bangchak issued me with a queue number for 2 weeks after my 365 day grace period expired. So I drove to DLT Cha-am (as I was passing by there at the time) and got the DL renewed painlessly there a few months ago with my Yellow Book with BKK address.
As a side note, it's strange that the Thai Embassy in Australia issued you with a Non B instead of a Non O visa, given that the documents required to get a Non B are more complex and not as common as with a Non O visa application.