It gets better.
I just completed semester 4 of 5 for my BSN and when I started, I had the exact same problems you did. I learned that for me, organization is key, and that my clinical instructor/nurse made a HUGE difference in my anxiety level.
For my organization, I have a piece of paper for each patient. Their vital information is all in the top left of the paper and I have spaces marked out for vitals, meds, treatments ordered, etc. so I can make sure everything gets done. I also STILL have to take my "assessment guide" in the room with me for shift assessments. When I'm practicing on classmates and students, I can do without, but I still haven't gotten over the anxiety enough to trust myself to remember everything I need to assess without a quick reminder. Don't be afraid to use your resources.
As far as anxiety, I learned that who I'm surrounded by makes a HUGE difference. With a clinical instructor focused on assessing my skills, I perform horribly. With a clinical instructor focused on helping you learn (like the one I had for med-surg) I performed much better. Also, nurses that acknowledge that I want to LEARN, not just perform skills, helped me alot. I loved when I was with a nurse who was willing to just let me follow him/her around. I didn't necessarily need them to slow down to let me perform a skill. I just needed them to let me observe how they did things and explain why they were doing what they did. That might just be my style of learning though. I am much more successful if I'm able to observe for quite awhile before diving in hands-on. You have to find what kind of learning style works best for you in a clinical setting since before now you've always been in a classroom setting, and things are very different.