Are the tests taken in your usual classroom? If not, spend some time studying in the room where the exams are held. Sometimes just being in a new location can add to the stress of exams. Being in a familiar environment also seems to help with recall.
If you have some sort of learning resource center on campus, see if they have additional tips that can help you relax more.
When you get the exam, first skim through the questions and jot down any initial ideas that come to mind for each question so you don't lose those thoughts as the exam progresses. Tackle the questions you are most confident about first; you don't have to work through the exam beginning to end, it's just fine to start somewhere in the middle and skip around. Save the toughest questions for last so you don't get bogged down and panicked when you get stuck on them and use your time first for the questions you have a good chance of getting right.
Can you solve all the problems on the exam that was returned now, without looking at the answer key? That will help tell you if it's really test anxiety or if you've missed learning something essential. There's a difference between having the book memorized and really understanding it and knowing how to use what you've learned. If you still can't solve the problems in a more relaxed environment, set up an appointment with your professor to find out how to improve your understanding of the material.
For some people arriving at the exam room early, finding your seat, and settling in with time to relax is helpful, for others that just gives them more time to dwell on the impending exam, and it's better to stay busy with something else in the few hours leading up to the exam and just set an alarm on your watch or cell phone to remind you when you need to get to the exam.
You should also pace your studying. Last minute cramming can increase anxiety. I used to find that sitting outside the room talking with other people before an exam made me more nervous, because their worrying about the exam and cramming minutes before the exam would get me worrying too, even if I knew the information they were cramming, so I'd just take a walk around the building or sit outdoors until it was time to enter the exam room.
Since you do well on quizzes, but not the exam, try to think of the exam as just a longer quiz...that really is all that it is.
If it's the exam environment itself that stresses you out...sometimes just the formality of having all the instructions read aloud, someone keeping time on the board, proctors roaming around hunting for cheaters, etc...then see if your professor is willing to give you the exam at a different time. Maybe just before the rest of the class takes it, you could take it in a more comfortable environment, such as a small conference room near his office. If you take the exam just before the rest of the class takes it, that'll show you're not trying to stall, or to get answers from other people before you take it, and there isn't enough time for you to get answers to them. There are other people who need to take exams individually as well due to things like learning disabilities or illness, so it's usually not difficult to accommodate one more student if they have a severe case of test anxiety that's hindering their performance.
