I have only entered a few cosplay contests for this very reason. I am very shy, no matter how well I know I've done on my costumes. I have been making costumes and clothing for ten years, and still have a bit of trouble telling people about them. I lucked out and the second time I entered costumes, my group and I won a Best Craftsmanship award for them. I felt like I did forget important things, but the girl that judged the Hall Cosplay Contest we entered was extremely nice, and as enthusiastic about techniques used as I was, so she made it very easy to talk to her.
I have had the experiences of doing lots of things on stages and in front of crowds. I used to sing in a choir and then I showed horses competitively until I was 18 (even showing at an invitation only world show), but still, none of those public experiences are the same as having someone scrutinize your handiwork. I just try to let my nerves stay down, and concentrate on what I have to say that will show what kind of work I have put into the costumes. It does help me that I have entered contests with at least one person with me. The first time I did, my friend had to make sure she was right next to me because I thought I would freak out and end up leaving the room before our turn came up. Support is something that's really important to have, even if you aren't competing. I know it helps me feel like I do better work than I give myself credit for at times.