Liz - no you are definitely not the only person auditioning for Elon from the West Coast
My D auditioned in October, and although we are from Northern California, the travel arrangements you have to make are essentially the same. Because of the time difference, and because there are no direct flights, you should plan on leaving by mid-day on Thursday, which implies an overnight stay. I checked the flights into GSO (Greensboro) from LAX and you have some options on Northwestern (via Memphis) and United via Chicago, both around $350 roundtrip. By the time we booked, the Acorn Inn was sold out and we stayed at the Country Suites.
From your post, it looks as if you may be traveling by yourself; if so, I'd recommend setting up a car service on beforehand like nydancemom mentioned. We rented a car, which is a lot cheaper, but this will not be possible if you are 18 years old.