Welcome! I also don't know much about Centrelink but I hope you don't mind talking about getting to a point where you can lead your own life in a fulfilling way.
I presented to hospital I think just before I joined this forum which looks like it was 4 years ago. I was suffering quite badly from depression and started to get quite bad anxiety too. Being in hospital took me out of work for a short while, and thankfully I didn't lose my job or anything. But I did lose a lot of friends and basically wasn't really living a fulfilling life for a very long time. What was interesting for me to realise in hindsight was that I'd probably been dealing with the same issues, just differently, since I was a child. So I am still seeing a psychologist now, 4 years on, and also the third psychologist, for those childhood issues. But the difference now is that, like you are trying to do, I've more or less learnt how to manage the worst of my mental health issues and being able to do this means I am living what I would consider a 'normal' life.
So I am not sure what the path forward will look for you, but I am really glad to hear that you are still trying to work with a psychologist to learn new skills. It sounds like you are able to do certain things in life like shopping or going out which is really positive to hear. I would suggest that holding that in mind, and remembering the good times, is going to be really helpful when you do start to worry about your future. With any luck, the good times will become more frequent and the bad times less so.
I hope you are able to work something out with Centrelink, but regardless, please feel free to reach out to us here if you just generally want to chat as well.