Many universities have a health center on campus. In some cases, the services there are free to students. They usually have counsellors on staff. You could check this out, because you may be able to see someone there and get some help for your depression and anxiety. Some also have a pharmacy with low cost drugs for students. Worth checking into.
If you were to transfer to another school now, with your health as it is, would you be able to do any better there than you're doing at this uni? If the answer is no, then be honest with yourself, and take some time away from school. Try to get a job, work for a while. Maybe go to school part-time, if you feel ready to do so. But don't just rush into another school if the reality is that by doing so, you're setting yourself up to fail.
If you can get a full time job that comes with health benefits, you can get treatment - and that may be something you want to try to do. You can return to school later, once you're well.
Per your question: if you're receiving financial aid at this uni, and you apply to another uni, they'll see this uni on your financial aid records, and you'll get caught.
But if you were to take some time off from school, deal with your issues, and start on your recovery; if you were perhaps to get a job during that time and be successful at it, then all that counts when you apply to your next university. You'd explain, in your essays, that you were ill at this uni (you don't have to go into details unless you want to), and your illness impacted your grades. That you've got that illness under control, and are now able to go back to school, and you're ready to succeed there. This is meaningful.