It doesn't surprise me to hear that your T-shirts lasted barely one wash. Like most other countries, imported souvenirs are plentiful and cheap ... some are cheap and not bad quality and some are cheap and nasty.
There are lots of "souvenir" places around Sydney - opposite the QVB building in George St springs to mind and I know there are various other shops along George Street heading down towards Circular Quay. There's also a large souvenir shop opposite CQ railway station.
It is those kinds of shops where you are most likely to find a collection of T-shirts with Aussies scenes or sayings on them.
For beach towels, I'd be looking at a large department store such as Myer or David Jones - you'll find them here - Market Street / Elizabeth St / George St
Alternatively, if you want to spend a little more on beach wear, head to The Rocks and peruse Ken Done's shop - http://www.kendone.com/contact/
If you're in the vicinity of one of the Australian Georgraphic shops, you might find some Aussie souvenirs there too - not necessarily beach towels but check out the store on line and see if there is anything you like - http://shop.australiangeographic.com.au/
Finally, The Rocks Markets on the weekends may have some hand crafted souvenirs but they're not likely to be beach towels etc.
Edited: 5:28 pm, October 26, 2012