I was like this at meets all throughout my gym career so I feel for you!
Visualization can really do wonders, if done consistently. You can go through your routine in your head wherever, and imagine yourself doing it perfectly. I would suggest doing it at practice too, even if you have no trouble there, so it takes the foreignness out of the competition setting a bit (you'll be doing the same thing [visualizing your routine before you do it] you do at practice which will tell your brain it's somewhat familiar with the situation).
If your coaches are willing, I also like to do pressure sets of routines with my gymnasts (as in we'll pretend we're at a meet, we'll get a coach with judge qualifications to sit next to the event looking
really serious
, have the rest of the group make unexpected noises (like stomping, clapping, laughing, shouting), play floor music in the background, set a timer etc.).