My wife is also from the US, and has had some similar issues living in Australia, which have fortunately improved. She had a lot of issues with some psychiatrists, both in the US and one in Australia, who were often doing more harm than good (such as putting her on medication that causes weight gain and then telling her she's fat, which she isn't at all).
Fortunately as she has improved, she no longer requires any medication, but has still been seeing a therapist who she met through the Beyond Blue program, and they have been amazing (I'm not sure if I'm allowed to recommend specific people or not, so I won't send their details unless the mods say it's all right). If you haven't already done it though, I highly recommend the Beyond Blue program in general, as it can be very helpful in many different ways.
Personally, I think the most important thing in a therapist is someone who genuinely cares and has empathy for people. Dealing with ones who barely seem to remember who you are each session and have no connection can really get to you. I wish I could give some secret formula for finding the perfect one, but unfortunately I think it's just trial and error. Any time you end up with someone who is a bad fit for you, don't hesitate to let them know it isn't working out and try someone else.
Lastly, make sure to give yourself enough credit. It's easy to think that you should be doing better, but you have already shifted to the other side of the world, which is a huge, huge change and achievement. It's good to try to keep on improving a little bit each day, but also take a bit of time each day to congratulate yourself on all the things you have done and are doing, no matter how small they may seem. For some people, just getting out of bed in the morning is a big thing, and if that's the case, they should congratulate themselves, and maybe they will even get something else done, and slowly grow to do more and more.
Or for other people, they might make $900k in a year and then beat themselves up because it wasn't a million. No matter how amazing it is to one person, it's a failure to another. At the same time, what may be a terrible failure to one person might be an amazing success to another.
Getting things organized financially in a new country can be extremely hard, and honestly my wife and I are in a similar situation. At the same time I think far, far more people are in the same situation than we often realize.