Hang in there, friend! The vet school application process was one of the most stressful periods of my life - it 100% paid off, but that doesn't negate how strenuous the entire process is....
The best thing I did to help my anxiety was to put my all into my application, so then once it was submitted, I could do my absolute best to step away and forget about it. Because for me I could tell myself that I had done everything possible to submit the best possible application I could, and the rest was up to the world. Plus, considering that I literally caught typos on my 5th or 6th read through that I had missed before (or had my spouse catch typos I'd missed on my own read throughs), the extra time to hone the application saved me stress in the long run.
The hardest part for me was the marathon and final sprint of finishing the application, only to have a waiting period of months that felt like torture. As much as you can, once you're finished, really try to step away and keep your mind busy - take up a hobby, do a new activity, spend extra time with friends. I wish I'd spent more time with friends, because I ended up choosing to move far away and I really miss them now. So, just try to live your life once everything's submitted - the time will pass and vet school applications are a true endurance test!
Do your best, submit a great application, then step away. Don't re-read your application because you WILL find typos. Because, it WILL cause you stress despite there being nothing you can do about it. Even with all the editing I did, I know that I found a few typos or general weird phrases I wish I'd corrected or altered. I remember sitting and biting my nails, worrying that calling Evans syndrome Evans disease would make a difference. And guess what? It didn't - I still was accepted to 8 schools.
Take time to take care of yourself, it can only help you - now, once you're in vet school, and once you're a practicing vet. You're in it for the long haul, so find those healthy coping mechanisms now!
You've got this!
ETA: An editing tip I learned in the working world was to take your final pieces (ALL of them, including experience sections), and go to a quiet place where you can read through everything - every single thing - OUT LOUD. Read it like you're reading it to someone. Your brain will catch typos that it would've skipped because you're forced to speak out loud. It's also a really great way to test for cadence and flow of your writing.