Can I cross the border to get a new 60-day visa for Thailand instead of flying back to Australia?

It can be believed that OP got his answer. Good luck with your stay and travel around.

Covid extension is still possible, just tick the box 'other' for the reasons, and I just noted: 'travelling' as the reason and it worked, this tip was given by a local employee here at the immigration

Vietnam is a great place to visit and obtain a tourist visa for Thailand.

Might I suggest this plan?

.. go to CW Thai immigration as planned today to get your 30 Day extension of stay...

... then before July 25th...go back to CW to apply for the 60 day COVID EXTENSION.

...that may be possible... BUT if denied you could be given 7 day stamp to leave Thailand...

...if that were to happen then follow up with your plan to go to Vietnam to apply for the Thai Tourist Visa.

... And IF you do get the COVID 60 day extension...you still could follow with your Vietnam Thai tourist visa trip after the 60 days COVID extension ends

Nope you can't cross the border and redo ANY visa 🙁

You BUY visas from thai consulates in other countries then come in on them, you can't get them just by exiting/reentering the country at the border

IF you're going to Vietnam for a week you could get a new single entry tourist visa at the thai consulate in Ho Chi Minh City OR Hanoi, (note HCMC requires you book an appointment to apply or you don't get in the gate)

Look at the website of the consulate you wanna use there for the requirements. Shouldn't have an issue applying for it if you meet the requirements. They only accept USD at the thai conuslates there and the single entry tourist visa is 40 USD