Staying some time in the Schengen area after your current visa expires is possible as the time under the work-holiday visa does not count toward the 90-day limit, see Does tourist visa (90 days) apply after a working holiday visa ends in Schengen countries?
Applying for a work-holiday visa from another country should also be no problem, at least as far as Schengen regulations are concerned. The basic principle is that long-stay visas are still national matters, not subject to the Schengen-wide restrictions on short stay (“Schengen visa” is a bit of a misnomer in this case, there are major differences between Schengen uniform short-stay visas and national long-term visas from Schengen countries).
If you are eligible under local law, a Schengen member state can certainly grant you a national visa starting immediately after another long-term national visa (from the same country or from another Schengen country). One thing you need to be mindful of is that it's sometimes impossible to apply for a long-stay visa from within the country (even if you would otherwise qualify for the visa) but some countries do allow it.
Importantly, none of this is true for short-stay visas (Schengen uniform type C visas) or, for Australian citizens, to visa-free short stays. In that case, the 90-day rule applies and it applies to you as a person. Getting another short-stay visa or using another passport does not entitle you to another 90 days in the Schengen area.
Finally, there is one last legal way to stay a bit longer in the Schengen area after having exhausted all work-holiday visas and visa-free short stays using prior bilateral agreements between Australia and specific Schengen member states. In particular, you should be able to get another 90 days in Denmark if your previous 90 days were spent elsewhere and possibly to stay longer in Germany as well.