Hey there, I had a journey similar to yours. Here's my advice.
First, you do not legally have to disclose any medical issue to your school, especially if you're not accepted and already attending a nursing program.
When my anxiety became a hindrance in the nursing program, the director did suggest I speak with disability services of the school. Since my issues were not really an issue related to school work, they couldn't do anything for me. However, it does sound like you may benefit from extended testing time. Be forewarned, they will require a direct proof of diagnoses from your psychiatrist, and likely will inform all your professors.
My main concern is that your anxiety and depression are not well-managed at this point. You said you are crying during every exam. That is rendering you barely functional. I hope that you have not accepted that as "your normal," and expect the world to bend around you. I am not saying this to be mean; I'm saying it because that's the real world. And if you want to be successful in the program and as a nurse, you MUST find a way to manage it successfully so that it doesn't have such a detriment on your work. Bursting into tears during every exam will quite simply prevent you from moving forward.
When is the last time you had your medications evaluated? I am asking this as someone who got through the program thanks to the graciousness of my professors and a horse's dose of zoloft. It was the first time in my life I had to swallow my pride due to mental health issues, and it wasn't easy. I kept getting my meds adjusted until I was functional enough to make it through. You need to keep looking for answers because the level you're at right now isn't functional.
As for the drug testing - most programs will only require one or two tests during the four semesters of courses, though of course students can be tested at any time. You should not be taking ativan daily, so my suggestion is, quit taking it a week before your drug test so you don't come up positive. The drug test is to rule out drug abusers, not find out every medication students are on. Be responsible and don't take ativan for at least 12 hours before your clinical days; if you can't sleep up to 10 mg of melatonin can be taken to help knock you out, recommended by my former psych.
Do not disclose your health status, mental or not, to anyone in the program unless it becomes enough of a problem to interfere with your success. If you are already expecting your mental health issues to be a huge issue during your schooling, then I think you're not ready yet and you need to get more stable before attending. School is stressful enough, combined with the fun stuff life throws at us, and you will see blood everywhere, people dying, open wounds, cursing, violence, etc., which are all enough to exacerbate mental health issues. You need to be on solid ground before you commit to school.
Good luck, and I hope you find the right path for you.