@pixel - I know I'm late in in this one, ignore this stuff if it's already been said.
Have you had your 10 free psychology appointments for the year? There's a lot of stuff that psychologists are no good for, but coping from day to day, especially with anxiety and depression related symptoms, they can be really good. Pick one connected to a larger practice, because they tend to have more information about other services in your city. You get a referral from your GP.
Which is suggestion 2. Have you got a good GP? They should be able to direct you to mental health services and programs, as well as an appropriate p-doc. Good gp's (not bulk billing) are worth their weight in gold with mental health help. They can also refer you for an additional free 5 psychology appointments (which is 15 every 12 months, more than once a month) if you have a particular illness (huge list that will qualify you: everything from diabetes to migraines, again, you need a good GP, forking out extra cash on your gp is worth it).
Third, take a look at the mental health hospitals websites for your cities. Most have day programs which are a good way in the back door, and if you have basic hospital health cover, many end up free.
Just some additional ideas. In a new city, you need to get your foot in the door, once you do, it comes a lot easier. Whatever public hospital catchment you're in will have an outpatient service, again, check the health dept websites. Getting in may require presenting to the ED for acute help, and you want to stay in at least overnight so that you get assigned a p-doc, rather than just seeing the one connected to the ED, although if they are out of beds (they'll tell you straight up usually), you could ask for a referral to the outpatient service to help you "stay in the community". Actually, that's just code for "I need to get into the system", but they shoumd assign you a psychologist or social worker in the community service which will be your makn support worker, and will help you get into the services you need.
Public hospitals are big (big big) on keeping people in the community. The quality varies a LOT between hospitals, but I received a lot of support this way in the past (had to try 3 different hospitals, which means 3 different catchments - use someone else's address, friend or relative, when you present to ED if you need to get into their catchment).
Hope some of that helps. Stick with it, help is out there. It takes patience, but you will get your foot in the door with some persistence:)