How can I control my anxiety, without the use of drugs?

Lots of good ways. Meditation, yoga, exercise, pilates, even weight training will help to reduce those anxiety levels. Healthy eating, valerian tea also helps.
Exercise. Exercise is a great way to relieve stress. Your body releases endorphins during exercise, which can help you feel calm. If you exercise 30 - 60 minutes a day, your stress levels can improve. When stressed, take 5 slow, deep breaths with your eyes closed, then roll your shoulders forward 5 times, then back 5 times. This will slow your heart rate and release tension in your neck and shoulders.