What you are experiencing is perfectly normal; in fact during a public speaking class I took I learned that performing or speaking to an audience was one of the top three things people have said they fear the most. Dying came in well behind it!
Everyone has heard the saying "Practice makes perfect" but it really should be "Perfect practice makes perfect". So, when practicing the parts where you have trouble, slow down the tempo a bit if you need to, and make sure you are technically correct and smooth; the speed will come on its own after you have the muscle memory (I play piano).
Also when you have a few free moments, visualize yourself playing the piece correctly, the audience applauding, etc. Performing well is just as much a mental as physical exercise and if you mentally prepare yourself to perform the piece you have a better chance of succeeding.
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