If you didn't extend your visa in country before the 60 days expired, then you are already on a severe overstay as 90 day tourist visas only grant 60 day stays within the 90 day visa validity period.
Look on the actual DTV group on Facebook for lots of first hand accounts of entries that both had and didn't have issues at various airports. This group doesn't allow me to tag another group so you'll have to search for it in Facebook groups.
Anonymous participant many airlines will ask to see the visa of you don't have an onward/return ticket booked within the 60 day visa exempt period. You have to prove how you will be able to enter Thailand legally for a longer stay.
It would be helpful to know what nationality you are, and what kind of previous entry history you have. If you had previously abused the visa exempt or education visa system to try to stay in Thailand long term, it would make sense for immigration officers to question the legitimacy of your DTV qualifications.
despite your best intentions, you were not ever supposed to apply for a visa from any embassy or consulate while you are still in Thailand. The evisa website specifically asks you your current location, as embassies/consulates require you to physically be in their service area (in your case, Australia) in order to grant you a visa. You got away with it, so do what everyone is telling you and exit and re-enter to activate that visa.
Assuming you incorrectly thought the 60 day tourist visa grants 90 day stays when it just means it can be used once to enter the country within 90 days, don't worry. Just go to immigration offices within Thailand and extend it for 30 days for a 1900 THB fee, giving you a 90 day total allowed stay.