I am always in amazement at how much luggage people want to drag around SE Asia..
Traveling light with the bare minimum makes life so much easier and more convenient..
And there is absolutely nothing you could possibly need that you cannot buy as you go should you need anything..
There are cheap laundry services available everywhere in the main tourist areas..
This is not a criticism but rather an observation and experience gained from dozens of trips all over SE Asia for many years..
It's not that difficult to travel light with the bare minimum if you are determined to do that..
Having said that..
If you have a reservation at a hotel for your return to Bangkok then often the hotel will store your bags while you are gone (assuming you are not going away for weeks at a time)
I don't have anything against Singapore itself except it's very extensive in comparison to other Asian countries and it's probably the most "un-Asian experience in SE Asia..
If that's what appeals to you then that's fine and enjoy..
I am just pointing out my personal observation from many year's of traveling all over SE Asia
Singapore is probably the most expensive destination you could choose for a short flight out of Thailand .
Do some research on Phnom Penh Cambodia as that's often a short cheap flight from Bangkok..
You will need a visa to enter Cambodia but visa on arrival is available for most passport countries or you can apply online say a week or so before you travel and save the hassle of lining up at the visa on arrival immigration desk..
Phnom Penh has quite a good nightlife and bar scene and it's relatively cheap for a short stay..
Vietnam could also be an option but depending on your passport country you may require a visa before you enter Vietnam (British passport holders for example don't require a visa for Vietnam and get 45 days entry on arrival) but many other countries require a visa prior to entering...
HCMC probably the best choice for a short visit and stay in District 1 near Bui Vien walking street for some nightlife and plenty of hotel options.
You can declare yourself a non resident for tax purposes.. your choice
But you would only do that in a situation where you have overseas income and paying tax in another country and don't want that income also subject to Australian taxation as well.